Download 1 Solution of Homework

Math 3181
Dr. Franz Rothe
February 4, 2011
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Solution of Homework
10 Problem 1.1 (Common tangents of two circles). How many common
tangents do two circles have. Informally draw all different cases, with 0, 1, 2, 3, 4 common tangents. Describe how they arise.
Answer. For any two different circles, there are five possibilities regarding their common
tangents:
(0) One circle lies inside the other. They have no common tangents.
(1) One circle touches the other from inside. There is one common tangent, located at
this touching point.
(2) The two circles intersect in two points. They have two common tangents, which lie
symmetrically to the axis connecting the two centers.
(3) The two circles touch each other from outside. They have three common tangents.
(4) The two circles lie outside of each other. They have four common tangents. These
are two pairs lying symmetrically to the axis connecting the two centers.
Figure 1: Two circles with no common tangents.
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Figure 2: Two circles with one common tangent.
Figure 3: Two circles with two common tangents.
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Figure 4: Two circles with three common tangents.
Figure 5: Two circles with four common tangents.
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10 Problem 1.2 (Common tangents of two circles). Given are two circles,
with center O and radius a, and center Q and radius b. They lie outside of each other.
Explain how to construct, with Euclidean tools, the common tangents of the two circles.
Actually do and describe the construction!
Figure 6: Construction of two of the four common tangents for two circles lying outside
each other.
The drawing on page 4 constructs just two common tangents, using construction 1.
The other two are their mirror images by line OQ connecting the centers.
Construction 1 (Common tangents of two circles). One constructs two new circles
around one of the given centers, say Q, choosing as their radii the sum a + b and
difference |a − b| of the radii a and b of two given circles. Next one constructs the
tangents from the second center O to these two new circles. Too, one needs the radii
where these tangents touch. The common tangents are produced by parallel shifts of these
tangents, to both sides, by a distance given by the radius of the original circle around O.
Shifting the tangents to the circle of radius a + b yields the inner common tangents,
whereas shifting the tangents to the circle of radius |a − b| results in the outer common
tangents.
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Figure 7: Halfing an angle with an isosceles triangle.
Proposition 1. Assume that the angle sum for every triangle is 2R. Then the base
angle of an isosceles triangle is half of the exterior angle at the top.
10 Problem 1.3. Prove proposition 1 from the assumptions that the angle sum
of any triangle is 2R, and the base angles of an isosceles triangle are congruent.
Answer (Reason for proposition 1). Let δ be the exterior angle at the top vertex A of
triangle 4ABC. Thus δ is the supplement of the interior angle α at that vertex. It was
assumed that the angle sum of any triangle is 2R. Hence
δ = 2R − α = α + β + γ − α = β + γ
which is the sum of the two nonadjacent interior angles. For an isosceles triangle with
top F = A, the two base angles β and γ are congruent by Euclid I.5. Hence the exterior
angle at the top is δ = 2β double the base angle. Hence the base angle β = 2δ is half of
the exterior angle δ.
Theorem 1 (Angles in a circle). The angle subtending any circular arc—with vertex
at the center of the circle—is twice the angle subtending the same arc with vertex on the
circle (Euclid III.20). Hence, if two angles inscribed in a circle subtend the same arc,
they are congruent (Euclid III.21).
We call the angle with vertex at the center central angle, and the angle with vertex
on the circle circumference angle of the given arc. In short, Euclid III.20 says:
The central angle is twice the circumference angle.
10 Problem 1.4. In the drawing on page 6 the circumference angle is γ = α + β
and the central angle is ω = x + y. Provide a drawing with appropriate notation for the
case that the circumference angle is γ = α − β and the central angle is ω = x − y. How
are points A, B, C, O located in this case?
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Figure 8: Central and circumference angle of a circular arc
Figure 9: Central and circumference angle of a circular arc, both obtained as differences.
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Answer. In the figure on page 6, the circumference angle is γ = α − β and the central
angle is ω = x − y. Since points A and O lie on different sides of line BC, central and
circumference angle are both obtained as differences.
10 Problem 1.5 (A construction using an altitude). Using Euclid III.21,
construct a triangle 4ABC with the three following pieces given: side c = AB = 6,
opposite angle γ = ∠BCA = 30◦ , and altitude hc = 4.
(hc is the altitude dropped from vertex C onto the opposite side AB).
(a) Do the construction and measure your angles α and β.
(b) Describe the steps for your construction.
Figure 10: A triangle construction
Answer.
Construction 2. Draw side AB = 6 and its perpendicular bisector p. Let M be the
midpoint of AB. The center O of the circum circle lies on the perpendicular bisector.
The center angle is double the circumference angle γ. Hence ∠AOB = 2γ = 60◦ , and
∠AOM = 30◦ . For the example given, the point O is especially easy to find because the
4AOB is equilateral. Next, we draw the circle around O through A and B. This is the
circum circle of 4ABC, on which vertex C lies.
Because of the given altitude hc = 4, vertex C lies on a parallel q to AB of distance
|M D| = hc . Finally, vertex C is an intersection point of this parallel with the circum
circle. One can choose any one of the two intersection points.
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10 Problem 1.6 (The midpoints of chords). Given is a circle C with center
O, and a point P inside C. Describe the location of the midpoints of all chords through
point P . Give a reason based on Thales’ Theorem or its converse.
Figure 11: Where lie the midpoints of all chords through point P ?
Answer. The midpoints of all chords in the given circle C through the point P lie on a
circle with diameter OP . Here is the reason: The midpoints are the foot points F of
the perpendiculars dropped from the center O of the circle onto the respective chords.
By the converse of Thales’ Theorem, all the vertices F of the right angles with sides
through the two points O and P lie on Thales’ circle with diameter OP .
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Figure 12: The midpoints of chords lie on Thales’ circle with diameter OP .
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Remark. Let the circles have center O and radius a, and center Q and radius b. These
cases correspond to:
(0) |OQ| < |b − a|: One circle lies inside the other.
(1) 0 < |OQ| = |b − a|: One circle touches the other from inside.
(2) |b − a| < |OQ| < a + b: The two circles intersect in two points.
(3) |b − a| < |OQ| = a + b: The two circles touch each other from outside.
(4) a + b < |OQ|: The two circles lie outside of each other.
(∞) 0 = |OQ| = |b − a|: The two circles are equal to each other.
Figure 13: Construction of a common tangent by a different method.
The drawing on page 10 constructs just one common tangent, with a different method
using similar triangles.
Construction 3 (A second way to get the common tangents of two circles). To
get the two outer common tangents, we begin by constructing their intersection point I.
We draw an arbitrary pair of parallel radii OS 0 and QT 0 , and get point I as intersection
of the lines S 0 T 0 and OQ. A special case occurs for two circles of equal radii—in that
case the lines S 0 T 0 , OQ and the common tangent ST , are all three parallel.
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To get the two inner common tangents, we draw a pair of any two anti-parallel radii
OS and QT 00 . The lines S 00 T 00 and OQ intersect in the intersection point J of the two
inner tangents.
Finally we construct the tangents from points I and J to anyone of the two circles,
and obtain their common tangents as well as the touching points.
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The idea behind the second construction. Because of symmetry, the intersection point
I lies on the line OQ through the two centers. Since the radii of the two circles to
their touching points S and T on one common tangent are parallel, we get equiangular
triangles 4IOS and 4IQT . By Thales’ (second) theorem, their sides are proportional:
|QI|
|OI|
=
|OS|
|QT |
The second pair of equiangular triangles 4XOS 0 and 4XQT 0 is obtained from an
arbitrary pair of parallel radii OS 0 and QT 0 and the intersection point X of the lines
S 0 T 0 and OQ. For these two equiangular triangles, we get now the proportions
|OX|
|QX|
|QX|
|OX|
=
=
=
|OS|
|OS 0 |
|QT 0 |
|QT |
Both proportions can only hold for X = I and hence we get the intersection point I.
Alternatively, we can use the Converse Desargues Theorem:
”If the sides of two triangles are pairwise parallel, then the two triangles are either
in perspective from a point, or the three lines through pairs of corresponding vertices
are parallel.”
Indeed, the two triangles 4OSS 0 and 4QT T 0 have pairwise parallel sides. Hence
they are either in perspective, or the three lines OQ, ST and S 0 T 0 are parallel.
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