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Math 348 - Topics in Geometry
Assignment 1
1. Trigonometric laws: Let triangle ABC have sides lengths a, b, c and angles α, β, γ as shown.
C
γ
a
b
A
β
α
c
B
Prove the following identities.
(a) Law of cosines:
(c) the area of triangle ABC satisfies
a2 = b2 + c2 − 2bc cos(α)
2 · | ABC | = bc sin(α)
(b) Law of sines:
(d) if R denotes the circumradius ABC,
sin α
sin β
sin γ
=
=
a
b
c
4R · | ABC | = abc.
2. The goal of this problem is to prove the following:
Theorem: Given an acute triangle ABC, the triangle formed by the centroids of the equilateral triangles constructed
on each of its sides is equilateral.
That is to say, that X, Y, Z in the following diagram form an equilateral triangle.
A
α
X
C
Z
c
b
γ
β
a
B
Y
(a) Prove that the 3 medians of a triangle meet at a point (the centroid). Deduce along with this that the
distance from the centroid to any vertex of a triangle is 23 the length of the corresponding median.
(b) Apply the cosine law and the result of part (a) to express the side length XZ (between centroids)
in terms of sides and angles of triangle ABC.
(c) You should notice a sum of angles in the previous part. Apply the first part of problem 1c and
cosine law (again) to express the length XZ (between centroids) in terms of only side lengths and
the area of triangle ABC.
(d) The previous formula implies the result stated at the beginning of this problem. Explain how.
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Math 348 - Topics in Geometry
Assignment 1
(e) All of this is fine and dandy, but there is another problem within this relating to the actual sketching
of such a diagram. That is, given a line segment AB with coordinates A = ( a1 , a2 ), B = (b1 , b2 ) in
the Euclidean plane, provide explicit coordinates for
i. a third point P that makes APB into an equilateral triangle and
ii. the centroid C of this equilateral triangle.
3. If the incenter and circumcenter of a triangle coincide, prove that the triangle must be equilateral.
4. Let ABC be an acute triangle. Let D be the point where the angle bisector of B intersects AC and E the
point where the angle bisector of C intersects AB. Let P and Q be the intersection points of DE and the
circumcircle of ABC. Show that AP = AQ if and only if AB = AC.
A
E
P
D
Q
β
α
B
β
α
C
5. Extend the median AX of triangle ABC until it meets the circumcircle of triangle ABC at K. There is a
unique circle which passes through A and B and is tangent to the line BC. If this circle meets AK at L,
prove that triangle LBX and triangle KCX are congruent to each other. Hence, deduce that BKCL is a
parallelogram.
A
L
X
B
C
K
6. Suppose that the circles C1 and C2 intersect at two points A and B and that C2 passes through the centre
of C1 , which we denote by O. Let X be a point on the arc AB which contains O and extend AX until it
intersects C1 at Y. Prove that XY = XB.
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Math 348 - Topics in Geometry
Assignment 1
X
A
Y
O
C2
B
C1
Hints
1. Drop an altitude.
2.
(a)
(b)
(c)
(d) This formula should admit some sort of symmetry...
(e) You may want to break into your linear algebra toolbox for this one
3. Draw in the line segments AI, BI, CI or equivalently, AO, BO, CO. What is special about the lengths in the diagram? What is
special about the angles in the diagram?
4. try to show that AP2 = AE · AB, do the same sort of thing with AQ. Equating these should suggest an important relation
between the triangles ABC and ADE. With this we can deduce that BEDC is cyclic. Now, how could we show that AB = AC?
5. Since the problem involves a tangent, you should try to use the alternate segment theorem. Use this to prove that ∠ LBX = ∠KCX.
6. How would you prove that triangle XYB is isosceles?
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