Download classwork geometry 5/13/2012

May 13, 2012
Math 9
Geometry.
Additional properties of ellipse, parabola and hyperbola
The optical property yields elementary proofs of some amazing results.
The isogonal property of conics. From any
point P outside an ellipse draw two tangents
to the ellipse, with tangency points X and Y.
Then the angles F1PX and F2PY are equal (F1
and F2 are the foci of the ellipse).
Proof. Let F1’, F2’ be the reflections of F1 and
F2 in PX and PY, respectively (see Figure).
Then PF1’ = PF1 and PF2’ = PF2. Moreover, the points F1, Y and F2 lie on a
line (because of the optical property). The same is true for the points
F2, X and F1’. Thus F2F1’ = F2X + XF1 = F2Y + YF1 = F2’F1. Thus, the
triangles PF2F1’ and PF1F2’ are equal (having three equal sides).
Therefore, ∠F2PF1 + 2∠F1PX =∠F2PF1’ = ∠F1P F2’= ∠F1PF2 + 2∠F2PY.
Hence ∠ F1PX =∠ F2PY, which is the desired result.
Consequence 1. The line F1P is the bisector
of the angle XF1Y.
A similar property holds for the hyperbola
Proof follows the same reasoning as for the
ellipse. Two cases should be considered
separately: when the tangency points X and
Y are either on different branches of the
hyperbola, or on the same branch, as in the
Figure.
Consequence 2. The locus of points from
which a given ellipse is seen at a right angle
(i.e., the tangents to the ellipse drawn from
such a point are perpendicular) is a circle
centered at the center of the ellipse (see
Figure).
Proof. Let F1, F2 be the foci of the ellipse and
suppose that the tangents to the ellipse at X
and Y intersect in P. Let F1’ be the reflection
of F1 in PX (see Figure). Then ∠XPY =∠F1’PF2 and F1’F2 = F1X+F2X, i.e.,
the length of the segment F1’F2 equals the major axis of the ellipse
(the length of the rope tying the goat). The angle F1’PF2 is right if and
only if |F1’P|2 +|F2P|2 = | F1’F2|2 (by the Pythagorean theorem).
Therefore XPY is a right angle if and only if |F1P|2 + |F2P|2 equals the
square of the major axis of the ellipse, (2a)2. But it is not difficult to
see that this condition defines a circle. Indeed, suppose F1 has
Cartesian coordinates (x1, y1), and F2 has coordinates (x2, y2). Then the
coordinates of the desired points P (x, y) satisfy the condition
But since the coefficients of x2 and y2 are equal (to 2) and the
coefficient of xy is zero, the set of points satisfying this condition is a
circle. By virtue of symmetry, its center is the midpoint of the
segment F1F2.
For the hyperbola such a circle does not always exist. When the angle
between the asymptotes of the hyperbola is acute, the radius of the
circle is imaginary. If the asymptotes are perpendicular, then the
circle degenerates into the point which is the center of the hyperbola.
The method of coordinates. Hyperbola.
Hyperbola can be defined as the locus of points
where the absolute value of the difference of
the distances to the two foci is a constant
equal to 2a, the distance between its two
vertices. This definition accounts for many of
the hyperbola's applications, such as
trilateration; this is the problem of determining
position from the difference in arrival times of
synchronized signals, as in GPS.
Similarly to the case of an ellipse, we write,
,
where (f,0) and (-f,0) are the positions of the
two foci, which we chose to lie on the X-axis.
Squaring the above equation twice, we arrive at
the canonical equation for the parabola,
, or,
, where
.
A hyperbola consists of two disconnected
curves called its arms or branches. At large
distances from the center, the hyperbola
approaches two lines, its asymptotes, which
intersect at the hyperbola's center. A
hyperbola approaches its asymptotes arbitrarily
closely as the distance from its center
increases, but it never intersects them;
however, a degenerate hyperbola consists only
The asymptotes of the hyperbola
(red curves) are shown as blue
dashed lines and intersect at the
center of the hyperbola, C. The two
focal points are labeled F1 and F2,
and the thin black line joining them
is the transverse axis. The
perpendicular thin black line
through the center is the
conjugate axis. The two thick black
lines parallel to the conjugate axis
(thus, perpendicular to the
transverse axis) are the two
directrices, D1 and D2. The
eccentricity e equals the ratio of
the distances from a point P on the
hyperbola to one focus and its
corresponding directrix line (shown
in green). The two vertices are
located on the transverse axis at
±a relative to the center. So the
parameters are: a — distance from
center C to either vertex;
b — length of a perpendicular
segment from each vertex to the
asymptotes; c — distance from
center C to either Focus point, F1
and F2, and θ — angle formed by
each asymptote with the
transverse axis.
of its asymptotes. Consistent with the symmetry of the hyperbola, if
the transverse axis is aligned with the x-axis of a Cartesian coordinate
system, the slopes of the asymptotes are equal in magnitude but
opposite in sign, ±b⁄a, where b=a×tan(θ) and where θ is the angle
between the transverse axis and either asymptote. The distance b (not
shown) is the length of the perpendicular segment from either vertex
to the asymptotes.
A conjugate axis of length 2b, corresponding
to the minor axis of an ellipse, is sometimes
drawn on the non-transverse principal axis;
its endpoints ±b lie on the minor axis at the
height of the asymptotes over/under the
hyperbola's vertices. Because of the minus
sign in some of the formulas below, it is also
called the imaginary axis of the hyperbola.
If b = a, the angle 2θ between the
asymptotes equals 90° and the hyperbola is
said to be rectangular or equilateral. In this
special case, the rectangle joining the four points on the asymptotes
directly above and below the vertices is a square, since the lengths of
its sides 2a = 2b.
If b = a, the angle 2θ between the asymptotes equals 90° and the
hyperbola is said to be rectangular or equilateral. In this special case,
the rectangle joining the four points on the asymptotes directly above
and below the vertices is a square, since the lengths of its sides 2a =
2b.
If the transverse axis of any hyperbola is aligned with the x-axis of a
Cartesian coordinate system and is centered on the origin, the equation
of the hyperbola can be written as
A hyperbola aligned in this way is called an "East-West opening
hyperbola". Likewise, a hyperbola with its transverse axis aligned with
the y-axis is called a "North-South opening hyperbola" and has equation
Every hyperbola is congruent to the origin-centered East-West opening
hyperbola sharing its same eccentricity ε (its shape, or degree of
"spread"), and is also congruent to the origin-centered North-South
opening hyperbola with identical eccentricity ε — that is, it can be
rotated so that it opens in the desired direction and can be translated
(rigidly moved in the plane) so that it is centered at the origin. For
convenience, hyperbolas are usually analyzed in terms of their centered
East-West opening form.
Other definitions of the hyperbola.
The above definition may also be expressed in terms of tangent circles.
The center of any circles externally tangent to two given circles lies on
a hyperbola, whose foci are the centers of the given circles and where
the vertex distance 2a equals the difference in radii of the two
circles. As a special case, one given circle may be a point located at one
focus; since a point may be considered as a circle of zero radius, the
other given circle—which is centered on the other focus—must have
radius 2a. This provides a simple technique for constructing a
hyperbola. It follows from this definition that a tangent line to the
hyperbola at a point P bisects the angle formed with the two foci, i.e.,
the angle F1P F2. Consequently, the feet of perpendiculars drawn from
each focus to such a tangent line lies on a circle of radius a that is
centered on the hyperbola's own center.
In stereometry, hyperbola can be defined
as a conic cross-section, similar to parabola
and ellipse. Namely, hyperbola is the curve
of intersection between a right circular
conical surface and a plane that cuts
through both halves of the cone. For the
other major types of conic sections, the
ellipse and the parabola, the plane cuts through only one half of the
double cone. If the plane is parallel to the axis of the double cone and
passes through its central apex, a degenerate hyperbola results that is
simply two straight lines that cross at the apex point.
Directrix and focus
Similarly to an ellipse, a hyperbola can be defined as the locus of points
for which the ratio of the distances to one focus and to a line (called
the directrix) is a constant ε. However, for a hyperbola it is larger
than 1. This constant is the eccentricity of the hyperbola. By symmetry
a hyperbola has two directrices, which are parallel to the conjugate
axis and are between it and the tangent to the hyperbola at a vertex.
Reciprocation of a circle*
The reciprocation of a circle B in a circle C always yields a conic section such as a
hyperbola. The process of "reciprocation in a circle C" consists of replacing every
line and point in a geometrical figure with their corresponding pole and polar,
respectively. The pole of a line is the inversion of its closest point to the circle C,
whereas the polar of a point is the converse, namely, a line whose closest point to C
is the inversion of the point.
The eccentricity of the conic section obtained by reciprocation is the ratio of the
distances between the two circles' centers to the radius r of reciprocation circle
C. If B and C represent the points at the centers of the corresponding circles,
then
Since the eccentricity of a hyperbola is always greater than one, the center B must
lie outside of the reciprocating circle C.
This definition implies that the hyperbola is both the locus of the poles of the
tangent lines to the circle B, as well as the envelope of the polar lines of the points
on B. Conversely, the circle B is the envelope of polars of points on the hyperbola,
and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B
have no (finite) poles because they pass through the center C of the reciprocation
circle C; the polars of the corresponding tangent points on B are the asymptotes of
the hyperbola. The two branches of the hyperbola correspond to the two parts of
the circle B that are separated by these tangent points.
Review the following problems from recent homework.
1.
A and B are given points, k is a given constant, SMAB is the area of
triangle MAB. Prove that the locus of points M, such that
, is a pair of straight lines.
2.
Find the equation of the locus of points equidistant from two
lines,
and
, where a, b, m, n are real
numbers.
Topical recap: vectors, translations.
A vector is a quantity that has magnitude (length) and
direction. This represents a translation. A vector is
represented by a directed line segment, which is a
segment with an arrow at one end indicating the
direction of movement. Unlike a ray, a directed line
segment has a specific length.
The direction is indicated by an arrow pointing from
the tail (the initial point) to the head (the terminal
point). If the tail is at point A and the head is at point
B, the vector from A to B is written as:
notation:
. Vectors may also be labeled as a single
letter with an arrow,
, or a single bold face
letter, such as vector v.
The length (magnitude) of a vector v is written |v|. Length is always a
non-negative real number.
C
Addition (subtraction?) of vectors.
One can formally define an operation of
addition on the set of all vectors (in the
space of vectors). For any two vectors,
and
, such an operation
results in a third vector,
, such that
three following rules hold,
A
B
C
A
B
(1)
(2)
(3)
D
Usually (but not necessarily always – not true for the so-called
“pseudo-vectors”), vector addition also satisfies the fourth property,
,
(4)
which allows to define the subtraction of vectors. Addition and
subtraction of vectors can be simply illustrated by considering the
consecutive translations. They allow decomposing any vector into a
sum, or a difference, of a number (two, or more) of other vectors.
The coordinate representation of vectors.
Let vector a define translation A->B under which an arbitrary point A
with coordinates (xA,yA) on the (X,Y) coordinate plane is displaced to a
point B with coordinates (xB,yB) = (xA+ax,yA+ay) on this plane. Then,
vector a is fully determined by the pair of numbers, (ax, ay), which
specify displacements along X and Y axis, respectively. If O is the
point of origin in the coordinate plane, (xO,yO) = (0, 0), it is clear, that
,
(5)
from where follows the coordinate notation of vectors,
= (xA,yA),
= (xB,yB), = (ax, ay) = a. Since numbers ax and ay denote magnitudes
of translation along the X and Y axes, respectively, corresponding to
the displacement by a vector a, it can be represented as a sum of
these two translations, a = ax ex + ay ey , or, (ax, ay) =(ax, 0) + (0, ay), or,
,
where ex =
and ey =
(6)
are vectors of length 1, called unit vectors.
The length (magnitude) of a vector, in coordinate representation, is
, a = |a| =
.
(7)
As you can see in the diagram at the right, the
length of a vector can be found by forming a
right triangle and utilizing the Pythagorean
Theorem or by using the Distance Formula.
The vector at the right translates 6 units to the
right and 4 units upward. The magnitude of the
vector is
from the Pythagorean Theorem, or
from the Distance Formula:
The direction of a vector is determined by the angle it makes with a
horizontal line. In the diagram at the right, to find the direction of the
vector (in degrees) we will utilize trigonometry. The tangent of the
angle formed by the vector and the horizontal line (the one drawn
parallel to the x-axis) is 4/6 (opposite/adjacent).
Scalar (dot) product of vectors.
One can formally define an operation of scalar multiplication on
vectors, consistent with the following definition of length, or
magnitude of a vector,
.
(8)
and the following properties, which hold if , , and
and r is a scalar.
are real vectors
C
 The dot product is commutative:
A
B
 The dot product is distributive over vector addition:
 The dot product is bilinear:
 When multiplied by a scalar value, dot product satisfies:
(these last two properties follow from the first two). Then, it is clear
from the drawing at the right, that
.
(9)
Two non-zero vectors a and b are orthogonal if and only if a • b = 0.
Unlike multiplication of ordinary numbers, where if ab = ac, then b
always equals c unless a is zero, the dot product does not obey the
cancellation law: if a • b = a • c and a ≠ 0, then we can write: a • (b − c)
= 0 by the distributive law; the result above says this just means that
a is perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore
b ≠ c.
The coordinate representation for the dot product is (a∙b) = axbx + ayby.
According to the definition given above, this must be equivalent to
(a∙b) = ab cos (
), which could be straightforwardly verified from
the drawing.
Cumulative recap: Math 9 reference material.
Optical property of ellipse, parabola and hyperbola
If a ray of light is reflected in a mirror, then the
reflection angle equals the incidence angle. This is
related to the Fermat principle, which states that the
light always travels along the shortest path. It is
clear from the Figure that of all reflection points P on
the line l (mirror) the shortest path between points F1
and F2 on the same side of it is such that points F1, P,
and the reflection of F2 in l, F’2, lie on a straight line.
The optical property of the ellipse. Suppose a
line l is tangent to an ellipse at a point P. Then l is
the bisector of the exterior angle F1PF2 (and its
perpendicular at point P is the bisector of F1PF2).
A light ray passing through one focus of an
elliptical mirror will pass through another focus
upon reflection.
Proof. Let X be an arbitrary point of l different from P. Since X is
outside the ellipse, we have XF1 +XF2 > PF1 +PF2, i.e., of all the points of
l the point P has the smallest sum of the distances to F1 and F2. This
means that the angles formed by the lines PF1 and PF2 with l are equal.
The optical property of the parabola. Suppose
a line l is tangent to a parabola at a point P. Let P’
be the projection of P to the directrix. Then l is
the bisector of the angle FPP’ (see figure).
If a point light source, such as a small light bulb, is
placed in the focus of a parabolic mirror, the
reflected light forms plane-parallel beam
(spotlights).
Proof. Let point P belong to a parabola and l’ be a bisector of the angle
FPP’, where |PP’| is the distance to the directrix l. Then, for any point
Q on l’, |FQ| = |QP’| ≥ |QQ’|. Hence, all points Q on l, except for Q = P,
are outside the parabola, so l’ is tangent to the parabola at point P.
Consider the following problem. Given two lines, l and l’,
P’
and a point F not on any of those lines, find a point P on
l
l such that the (signed) difference of distances from
it to l’ and F is maximal (#2 in the homework). As seen
L’
l’
in the figure, for any P’ on l the distance to l’, |P’L’| ≤
|P’L| ≤ |P’F| + |FL|, where |FL| is the distance from F to l’. Hence, |P’L’|
- |P’F| ≤ |FL| and the difference is largest (=|FL|) when point P belongs
to the perpendicular FL from point F to l’.
The optical property of the hyperbola. Suppose a
line l is tangent to a hyperbola at a point P; then l is
the bisector of the angle F1PF2, where F1 and F2 are
the foci of the hyperbola (see figure).
Proof. Let point P belong to a hyperbola and l’ be a
bisector of the angle F1PF2. Let F1’ be the reflection of
F1 in l’. Then, for any point Q on l’, |QF1| = |QF1’|, and
|QF2| - |QF1| = |QF2| - |QF1’| ≤ |F2F1’|, again by the triangle inequality.
Hence, all points Q on l’, except for Q = P, are in-between the branches
of the hyperbola, so l’ is tangent to the hyperbola at point P.
Consider the following problem. Given line l and points F1
and F2 lying on different sides of it, find point P on the
line l such that the absolute value of the difference in
distances from P to points F1 and F2 is maximal (#1 in
the homework). As above, let F2’ be the reflection of F2
in l. Then for any point X on l, |XF1| - |XF2| ≤ |F1F2’|.
P
F
L
Optical property of ellipse, parabola and hyperbola (continued).
If a light source is placed at one focus of an
elliptic mirror, all light rays on the plane of the
ellipse are reflected to the second focus. Since
no other smooth curve has such a property, it
can be used as an alternative definition of an
ellipse. (In the special case of a circle with a
source at its center all light would be reflected
back to the center.) If the ellipse is rotated
along its major axis to produce an ellipsoidal mirror (specifically, a
prolate spheroid), this property will hold for all rays out of the source.
Alternatively, a cylindrical mirror with elliptical cross-section can be
used to focus light from a linear fluorescent
lamp along a line of the paper; such mirrors are
used in some document scanners. 3D elliptical
mirrors are used in the floating zone furnaces
to obtain locally high temperature needed for
melting of the material for the crystal growth.
Sound waves are reflected in a similar way, so in
a large elliptical room a person standing at one
focus can hear a person standing at the other focus remarkably well.
In the 17th century, Johannes Kepler discovered that the orbits along
which the planets travel around the Sun are ellipses with the Sun at
one focus, in his first law of planetary motion.
The method of coordinates. Ellipse.
An ellipse is a smooth closed curve which is
symmetric about its horizontal and vertical axes.
The distance between antipodal points on the
ellipse, or pairs of points whose midpoint is at the
center of the ellipse, is maximum along the major
axis or transverse diameter, and a minimum along
the perpendicular minor axis or conjugate
diameter.The semi-major axis (denoted by a in
the figure) and the semi-minor axis (denoted by b in the figure) are
one half of the major and minor axes, respectively. These are
sometimes called (especially in technical fields) the major and minor
semi-axes, the major and minor semiaxes, or major radius and minor
radius.
The foci of the ellipse are two
special points F1 and F2 on the
ellipse's major axis and are
equidistant from the center point.
The sum of the distances from
any point P on the ellipse to those
two foci is constant and equal to
the major axis ( PF1 + PF2 = 2a ).
Each of these two points is called
a focus of the ellipse.
Excersise. Prove that the sum of the distances from any point inside
the ellipse to the foci is less — and from any point outside the ellipse is
greater — than the length of the major axis.
Consider the locus of points such that the sum of the distances to two
given points A(-f,0) and B(f,0) is the same for all points P(x,y). It is an
ellipse with the foci A and B. This distance is equal to the length of the
major axis of the ellipse, 2a. Then, for every point on the ellipse,
,
Where from, by squaring this equation twice, we obtain for an ellipse,
.
The equation of an ellipse whose major and minor axes coincide with
the Cartesian axes is
. The area enclosed by an ellipse is πab,
where (as before) a and b are one-half of the ellipse's major and minor
axes respectively.
The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of
the distance between the two foci, to the length of the major axis or e
= 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0 < e <
1). When it is 0 the foci coincide with the center point and the figure
is a circle. As the eccentricity tends toward 1, the ellipse gets a more
elongated shape. It tends towards a line segment if the two foci
remain a finite distance apart and a parabola if one focus is kept fixed
as the other is allowed to move arbitrarily far away. The distance ae
from a focal point to the centre is called the linear eccentricity of the
ellipse (f = ae).
Other definitions of the ellipse.
Each focus F of the ellipse is associated with a line parallel to the
minor axis called a directrix. Refer to the figure. The distance from
any point P on the ellipse to the focus F is a constant fraction of that
point's perpendicular distance to the directrix resulting in the equality,
e =PF/PD < 1.
In stereometry, an ellipse is defined as a plane curve
that results from the intersection of a cone by a plane
in a way that produces a closed curve. Circles are
special cases of ellipses, obtained when the cutting
plane is orthogonal to the cone's axis.
Consider a circle with the center O, which is internally
tangent to two given circles with centers F1 and F2
and radii R and r, respectively, such that one is
inside the other (see figure). The sum of the
distances from O to the centers F1 and F2, equals
|OF1| + |O F2| = R + r, and is independent of the
position of the circle (O,r’). Therefore, the locus
of centers of all such circles [internally tangent to
two given nested circles (F1,R) and (F2,r)] is an
ellipse with foci F1 and F2.
Y
O
r’
r’
R
F2
F1
0
r
X
Y
Consider circles (F1,R) and (F2,r) that are not
nested. Then the loci of the centers O of circles
externally tangent to these two satisfy |OF1| - |O
F2| = R + r.
F2
F1
0
r
X
R
r’
r’
O
The method of coordinates. Parabola.
Consider locus of points equidistant from a given
point F and a given line l, which does not contain
this point. It is a parabola with focus F and the
directrix l. The line perpendicular to the directrix
and passing through the focus (that is, the line that
splits the parabola through the middle) is called the
"axis of symmetry". The point on the axis of
symmetry that intersects the parabola is called the
Y
y = kx2
(0,a)
X
O
(0,-a)
y = -a
"vertex", and it is the point where the curvature is greatest.
The easiest way to show that the above definition indeed corresponds
to a parabola, is by using the method of coordinates. Indeed, let line l
be parallel to the X-axis and intersect the Y-axis at y = -a, and focus F
(0, a) lie on the Y-axis at the same distance a from the origin. Then for
any point on the parabola according to the definition above,
, wherefrom
.
If the directrix is a line y = a, and the focus has
some general coordinates, F(xF, yF), then the points
on the parabola satisfy the equation
, where
and
. If the
Y
y = k(x-xF)2+b
directrix is parallel to the Y-axis, then parabola’s
equation becomes x = k (y - yF)2+b.
(xF,yF)
Parabolas can open up, down, left, right, or in some
other arbitrary direction. Any parabola can be
repositioned and rescaled to fit exactly on any
other parabola — that is, all parabolas are similar.
How do you think the equation of a parabola with
the directrix y = ax + b, at an arbitrary angle
atan(a) to the (X,Y) axes, looks like? In order to
solve this problem, we need to learn how to find a
distance from the point P(x,y) to a line y = ax + b.
Alternatively (in stereometry), parabola is defined
as a conic section, similar to the ellipse and
hyperbola. Parabola is a unique conic section,
created from the intersection of a right circular
conical surface and a plane parallel to a generating
straight line of that surface. The parabola has
many important applications, from automobile
y=a
(0,a)
X
O
Y
(xF,yF)
y=
ax
+b
(0,b)
O (-B/a,0)
X
headlight reflectors to the design of ballistic missiles. They are
frequently used in physics, engineering, and many other areas.
The method of coordinates.
In introducing coordinates, we set up a correspondence between
numbers and points on a straight line. The following property is
satisfied: to each point on the line there corresponds one and only one
number, and to each number there corresponds one and only one point.
C
D
A
B
C
D
A
B
O
A correspondence between two sets where for each element of the
first set there is one element in the second set, and each element in
the second set corresponds to some element of the first set is called a
one-to-one correspondence. Can there be a one-to-one correspondence
between two line segments of unequal length (see Figure above)?
Which set of points is defined by the following
relation (draw it on the coordinate plane)?
Y
II
I
a. |x| = |y|
|y| = |x|
b. |x|+x = |y|+y
Y
X
O
c. |x|/x = |y|/y
d. x2 – y2 < 0
III
IV
e. x2 + y2 > 1
O
X
f. x2 + 8x = 9-y2
g. [y] = [x]
h. {y} = {x}
The radical axis of two circles.
For any circle of radius R and any point P distant d from the center,
the quantity d2-R2 is called the power of P with respect to the circle.
Consider the locus of
all points whose powers
with respect to two
non-concentric circles
are equal. These points
form a straight line
perpendicular to the
line of centers of the
two circles. This line is
called the radical axis
of two circles.
Y
(x,y)
T
T’
r
r’ O’
(a’,0)
0
O
(a,0)
The easiest proof is by writing the relation for coordinates (x,y) of
such points (Coxeter, Gretzer, pp. 31-33),
.
This defines line x = const, which is perpendicular to the X-axis.
Proof of the necessary condition of Ptolemey’s theorem, i.e. of Eq. (1)
for an inscribed quadrilateral.
X
Geometrical proof employs an elegant supplementary
C
b
construct. Inventing such an additional

B 
 
geometrical element is one of the key, most
d1
important and powerful methods of geometrical a  P


proof.
c

A
d2
E

Draw segment CE, whose endpoint, E, belongs to
the diagonal BD, and which s at an angle
to
the side CD. Thus obtained DEC ~ ABC. Therefore,
d


D
.
Furthermore,
and therefore BCE ~ ACD, so
.
Adding thus obtained equalities, we get
ac+bd = |AC||ED| + |AC||BE| = d1d2.
The sufficiency of this condition is easy to prove by contradiction.
Application of Euclid’s theorems: Eulers’ formula.
Using the Euclid’s theorem on chords in the circle, the
following formula for the distance between the
incenter and the circumcenter of a triangle can be
established.
Let O and L be the circumcenter and incenter
B
(that is, centers of the circumscribed and the
inscribe circle), respectively, of a triangle ABC
with circumradius R and inradius r. Then the distance
|OL| = d is given by
M
A
B’
R
r L
r d O
R
A’
d2 = R2 – 2Rr.
Using Euclid’s formula for the chords AA’ and BB’, we get |AL||LA’| =
|BL||LB’| = R2-d2. From rectangular ,
. Similarly,
C
from rectangular A’MB based on diameter A’M,
. Now, because L is the center of
the inscribed circle,
and
, so
. This means that A’MB is
equilateral, so
, wherefrom
, which proves the Euler’s formula.
Properties of inscribed quadrilaterals. Ptolemey’s theorem.
Consider the quadrilateral ABCD inscribed into a
circle. It is clear from the theorem on the
B 
inscribed circle that the opposite angles of
a 
ABCD are supplementary (i. e. add to 180 deg.),


i.
A
b
P
C


 d1
  d2
d
 
A quadrilateral can be inscribed in a circle
if and only if its opposite angles are
supplementary.
Now consider angles , , , , between the sides and the diagonals. The
angle between the diagonals,  - ( + ) =  + .
ii.
By summing up areas of the four triangles, APB, BPC, CPD and
DPA, obtained by drawing the diagonals AC and BD, we obtain
the area of the quadrilateral,
SABCD = ½ d1d2 sin()
iii.
Ptolemey’s theorem. A quadrilateral can be inscribed in a circle
if and only if the product f its diagonals its equals the sum of
the products of its opposite sides,
d1d2 = ac+bd
(1)
c
D
Euclids’ theorems. Power of point with respect to the circle.
1. Consider the following figures. Using the
theorem on the angle inscribed into a
circle and the similarity of the
corresponding triangles, it is easy to
prove the following Euclid theorems.
i. If two chords AC and BD intersect at
a point P inside the circle, then
|AP||PC| = |BP||PD|= R2-d2,
where R is the radius of the circle and
and d is the distance from point P to the
center of the circle, d=|PO|.
ii.
If two chords AD and BC intersect at
a point P’ outside the circle, then
|P’A||P’D| = |P’B||P’C|=|PT|2= d2-R2,
where |PT| is a segment tangent to the circle.
For any circle of radius R and any point P distant d from the center,
the quantity d2-R2 is called the power of P with respect to the circle.
Vectors. Translations.
A vector is a quantity that has magnitude (length)
and direction. This represents a translation.
A vector is represented by a directed line segment,
which is a segment with an arrow at one end
indicating the direction of movement. Unlike a ray, a
directed line segment has a specific length.
The direction is indicated by an arrow pointing from
the tail (the initial point) to the head (the terminal
point). If the tail is at point A and the head is at
point B, the vector from A to B is written as:
notation:
. Vectors may also be labeled as a single
letter with an arrow,
, or a single bold face
letter, such as vector v.
The length (magnitude) of a vector v is written |v|.
Length is always a non-negative real
number.
Addition (subtraction?) of vectors.
One can formally define an operation of
addition on the set of all vectors (in the
space of vectors). For any two vectors,
and
, such an operation
results in a third vector,
, such
that three following rules hold,
A
C
B
C
A
D
B
(1)
(2)
(3)
Usually (but not necessarily always – not true for the so-called
“pseudo-vectors”), vector addition also satisfies the fourth property,
,
(4)
which allows to define the subtraction of vectors. Addition and
subtraction of vectors can be simply illustrated by considering the
consecutive translations. They allow decomposing any vector into a
sum, or a difference, of a number (two, or more) of other vectors.
The coordinate representation of vectors.
Let vector a define translation A->B under which an arbitrary point A
with coordinates (xA,yA) on the (X,Y) coordinate plane is displaced to a
point B with coordinates (xB,yB) = (xA+ax,yA+ay) on this plane. Then,
vector a is fully determined by the pair of numbers, (ax, ay), which
specify displacements along X and Y axis, respectively. If O is the
point of origin in the coordinate plane, (xO,yO) = (0, 0), it is clear, that
,
(5)
from where follows the coordinate notation of vectors,
= (xA,yA),
= (xB,yB), = (ax, ay) = a. Since numbers ax and ay denote magnitudes
of translation along the X and Y axes, respectively, corresponding to
the displacement by a vector a, it can be represented as a sum of
these two translations, a = ax ex + ay ey , or, (ax, ay) =(ax, 0) + (0, ay), or,
,
where ex =
and ey =
(6)
are vectors of length 1, called unit vectors.
The length (magnitude) of a vector, in coordinate
representation, is
, a = |a| =
.
(7)
As you can see in the diagram at the right, the
length of a vector can be found by forming a right
triangle and utilizing the Pythagorean Theorem or
by using the Distance Formula.
The vector at the right translates 6 units to the right and 4 units
upward. The magnitude of the vector is
from the Pythagorean
Theorem, or from the Distance Formula:
The direction of a vector is determined by the
angle it makes with a horizontal line.
In the diagram at the right, to find the direction
of the vector (in degrees) we will utilize
trigonometry. The tangent of the angle formed by
the vector and the horizontal line (the one drawn
parallel to the x-axis) is 4/6 (opposite/adjacent).
Scalar (dot) product of vectors.
One can formally define an operation of scalar multiplication on
vectors, consistent with the following definition of length, or
magnitude of a vector,
.
(8)
and the following properties, which hold if , , and
and r is a scalar.
are real vectors
C
 The dot product is commutative:
A
B
 The dot product is distributive over vector addition:
 The dot product is bilinear:
 When multiplied by a scalar value, dot product satisfies:
(these last two properties follow from the first two). Then, it is clear
from the drawing at the right, that
.
(9)
Two non-zero vectors a and b are orthogonal if and only if a • b = 0.
Unlike multiplication of ordinary numbers, where if ab = ac, then b
always equals c unless a is zero, the dot product does not obey the
cancellation law:
If a • b = a • c and a ≠ 0, then we can write: a • (b − c) = 0 by the
distributive law; the result above says this just means that a is
perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore b ≠
c.
The coordinate representation for the dot product is (a∙b) = axbx + ayby.
According to the definition given above, this must be equivalent to
(a∙b) = ab cos (
), which could be straightforwardly verified from
the drawing.
Recap on addition operation.
Addition is a mathematical operation that represents combining
collections of objects together into a larger collection. It is signified
by the plus sign (+). A nice review on various aspects of addition
operation is on http://en.wikipedia.org/wiki/Addition.
Addition obeys several important laws. It is
 commutative, meaning that order does not matter
 associative, for the number addition this means that when one
adds more than two numbers, order in which addition is
performed does not matter.
 addition of zero, denoted 0, does not change the result; for
numbers repeated addition of 1 is the same as counting.
Translations and vectors.
The translation at the left shows a vector
translating the top triangle 4 units to the
right and 9 units downward. The notation
for such vector movement may be written
as:
or
Vectors such as those used in translations
are sometimes called “free vectors”. A “free” vector is an infinite set
of parallel directed line segments and can be thought of as a
translation. Any two vectors of the same length and parallel to each
other are considered identical. They need not have the same initial and
terminal points. Notice that the vectors in the translation which
connect the pre-image vertices to the image vertices are all parallel
and are all the same length.
You may also hear the terms "displacement" vector or "translation"
vector when working with translations.
Position vector: To each “free” vector (or
translation), there corresponds a position vector
which is the image of the origin under that
translation.
Unlike a “free” vector, a position vector is "tied" or
"fixed" to the origin. A position vector describes
the spatial position of a point relative to the origin
The Law of Cosines.
For any triangle ABC,
(1)
To prove it, we consider right triangles formed by
the height AM,
,
,
B

a
c
,

A
h
M

b
Relations in the right triangle. Tangent and
cotangent.
In the right triangle, a leg is equal to the
hypotenuse times sine of the angle opposite to that A 
C
B
c
a
b
C
leg,
to it,
, or to a hypotenuse times cosine of the angle adjacent
.
In the right triangle, a leg is equal to the other leg times tangent of
the angle opposite to it (the first leg),
, or to the other leg
times cotangent of the angle adjacent to it,
.
Tangent and cotangent are therefore defined as,
(1)
Inverse trigonometric functions:
,
.
If we are given an angle, 30 for example, then we can find
Inversely, if we are given a value of the sine function, say, ½, then the
challenge is to name the angle x, such that
"The sine of what angle is equal to ½?"
The Extended Law of Sines
B



a
c


The extended Law of Sines states that for any
triangle ABC,

C
b
A
(2)
where R is the radius of the circumscribed circle.
The Law of Sines states the above without including the last equality
to 2R and generalizes the fact that the greater side lies opposite to
the greater angle.
The Law of Sines (another proof).
B
For a triangle ABC


h2
c


A
a
h1
M
The Law of Cosines.
For any triangle ABC,
(3)

b
C
To prove it, we consider right triangles formed by
the height AM,
,
,
B
a
c
,

A
h
M
C
b
Formulas for the area of a triangle.
Using the Law of Sines in a standard formula, or
simply considering the right triangle formed by an
altitude opposite to vertex A, one obtains,
B

a
c
h

Similarly, we also can get two more formulas:

b
A
Using the Law of sines, we also have,
where R is the radius of the circumscribed circle. We have also
previously shown that
C
where r is the radius of the inscribed circle and s the semiperimeter.
Finally, the area of the triangle can also be derived from the lengths of
the sides by the Heron’s formula,
Definition of sinα and cosα.
For any acute angle
includes angle .
B
we draw a right triangle that
c

A
The sine of is a ratio of the length of the leg
opposite this angle to the length of the hypotenuse of the triangle.
a
b
The cosine of is a ratio of the length of the leg adjacent to this
angle to the length of the hypotenuse of the triangle.
Tangent and cotangent are
1. Theorem
The values of the trigonometric ratios of an acute angle depend only
on the size of the angle itself, and not on the particular right
triangle containing the angle.
2. Show that
3.
4. Compute
, but
,
.
C
5. Fill in the following table (first row can be filled
with the help of the triangle):
1
a

6.
, if
are complementary angles.
7. If a and b are pair of numbers such that
, then there
exist an angle such that
.
8. Inverse trigonometric functions:
,
.
Inscribed and excribed circles. Extended Law of Sines.
If is the angle subtended by a chord BC with
the vertex at a point A on the circle of radius
R, then
. For a triangle ABC with the
sides a, b, c,
where R is a radius of excribed circle.
The Rowland circle.
A
A
M
M
B
B
C
S
S
C
9. The inscribed angle theorem.
The inscribed angle theorem states that an angle
α inscribed in a circle is half of the central angle
β=2α that subtends the same arc on the circle
(Fig.1) or complete half of it to 180. Therefore,
the angle does not change as its apex is moved to
different positions on the circle.
Proof.
First, let us deal with the simple case when one
of the rays of angle ACB passes through the
center of the circle (Fig. 2). AOB (β) is a
central angle that subtends the same arc as
ACB (α).
Triangle AOC is an isosceles triangle because
|OA|=|OC|, so angle OAC and angle OCA are
equal and angle OC=180 - 2α, but it is also
equal 180-β as a supplement angel to angle β.
AOC=180 – 2α =180-β  β = 2α.
Fig.2
In the case when center of the circle placed inside of angle ACB we can
divide the angle ACB with a ray CB' passing through the center of the
circle. Now we have two inscribed angles:
C
angle ACB' and angle B'CB, each of them has

one side which passes through the center of
the circle and can use previous part to proof
O
B
that β=2α.

α = ϕ + ϕ',
A
B’
C

β = ψ + ψ' = 2ϕ + 2ϕ' = 2(ϕ + ϕ') = 2α.
B
O
When center of the circle is outside of inscribed
angle, we can draw a ray from a vertex of our angle
through the center the circle. Then angle
ACB()='CB(ϕ') - 'CA(ϕ) and we again can use the
first part.

A
B’
β = ψ'- ψ= 2ϕ'- 2ϕ= 2(ϕ' - ϕ) = 2α.
Only the case of obtuse angle is left. In this case
the ray CB' passes through the center of the circle
and divides angle ACB into two angles ϕ and ϕ'
They are not now half of the angles ψ and ψ', but
half of their supplement angles χ and χ' therefore
B’

’
O
B


A
C ½
1
1
1
1
1
1
α = χ+ χ' = (χ+ χ ')= (180−ψ+ 180−ψ')=180− (ψ+ ψ')=180− β
2
2
2
2
2
2
“Direct” and “Inverse” Theorems.
Each theorem consists of premise and conclusion. Premise is a
proposition supporting, or helping to support a conclusion.
If we have two propositions, A (premise) and B (conclusion), then we
can make a proposition A  B (If A is truth, then B is also truth, A
is sufficient for B, or B follows from A). This statement is
sometimes called the “direct” theorem and has to be proven.
Or, we can construct a proposition A  B (A is truth only if B is also
truth, B is necessary for A, or A follows from B), which is sometimes
called the “inverse” theorem, and also has to be proven.
While some theorems offer only necessary or only sufficient
condition, many theorems establish equivalence of two propositions,
AB.
10.
“Inverse” Ceva’s theorem. Geometrical proof.
For the Ceva’s theorem the premise (A)
is “Three chvians in a triangle ABC,
AA’, CC’, BB’, are concurrent”. The
conclusion (B) is,
B
C’
A
O
A’
B’
C
“
”. The full
statement of the “direct” theorem is A
 B, i. e.,
If three chevians in a triangle ABC, AA’, CC’, BB’, are concurrent, then
is true. From A follows B (A b). We proved this
last time in class. Again, premise in the “direct” theorem provides
sufficient condition for the conclusion to be truth.
Let us formulate the “inverse Ceva’s theorem”, the theorem where
premise and conclusion switch places.
If in a triangle ABC three chevians divide sides in such a way that
holds, then they are concurrent. A follows from B, BA, or AB, or,
~A~B, in other words if the three chevians of a triangle ABC are not
concurrent , then
. Three chevians being
concurrent is a necessary condition for the relation
to hold.
In the case of Ceva’s theorem premise and conclusion (propositions A
and B) are equivalent, (AB), and we can state the theorem as follows
Ceva’s theorem. Three chevians in a triangle ABC, AA’, CC’, BB’, are
concurrent, if and only if
11.
.
“Inverse” Thales theorem.
The “inverse” Thales theorem states
C
A
If lengths of segments in the Figure on
C’
the left satisfy
B
B’
, then lines
BC and BC’ are parallel. The proof is
similar to the proof of Ceva’s “inverse”
theorem.
If a theorem establishes the equivalence of two propositions A and B,
AB, it is actually often the case that the proof of the necessary
condition, AB, i. e. the “inverse” theorem, is much simpler than the
proof of the “direct” proposition, establishing the sufficiency, A  B.
It often could be achieved by using the sufficiency condition which has
already been proven, and employing the method of “proof by
contradiction”, or another similar construct.
Examples of necessary and sufficient statements
 Predicate A: “quadrilateral is a square”
Predicate B: “all four its sides are equal”
Which of the following holds: A  B, A  B, A  B?
Is A necessary or sufficient condition for B?
If a quadrilateral is not square its four sides are not equal. Truth
or not? (AB or ~A~B).
 Predicate A:
Predicate B:
Which of the following holds: A  B, A  B, A  B?