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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, AB. 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, BA, or AB, 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, (AB), 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, AB, it is actually often the case that the proof of the necessary condition, AB, 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? (AB or ~A~B). Predicate A: Predicate B: Which of the following holds: A B, A B, A B?