SECTION 1.2 Triangle Geometry 27
cosθ =
b^2 −s^2 −p^2
2 ps.
Equating the two expressions and noting thata=r+seventually leads
to the desired result.
Corollary [Apollonius Theo-
rem]. We are given the triangle
4 ABC, with sides a, b,and c, to-
gether with the medianBX, as in-
dicated in the figure to the right.
Then
b^2 +c^2 = 2m^2 +a^2 / 2.If b = c (the triangle is isosceles),
then the above reduces to
m^2 + (a/2)^2 =b^2.This follows instantly from Stewart’s Theorem.
Exercises
- Assume that the sides of a triangle are 4, 5, and 6.
(a) Compute the area of this triangle.
(b) Show that one of the angles is twice one of the other angles.- (The Golden Triangle)You are
given the triangle depicted to the
right with 4 ABD∼ 4BCAShow
that
DC
AD
=
√
5 + 1
2
, the golden
ratio.