Advanced High-School Mathematics

26 CHAPTER 1 Advanced Euclidean Geometry

4. In the triangle to the right, show
   that c =

√

1 +i+

√

1 −i

√ 42 (where

i^2 =−1)

1 1

c

135 ◦

5. Given 4 ABCwithCa right angle, letDbe the midpoint of [AB]
   and show that 4 ADC is isosceles withAD=DC.


6. Given 4 ABCwithBC =a, CA=b, andAB=c. LetDbe the
   midpoint of [BC] and show thatAD=^12

»

2(b^2 +c^2 )−a^2.

1.2.7 Algebraic results; Stewart’s theorem and Apollonius’
theorem

Stewart’s Theorem. We are
given the triangle 4 ABC, together
with the edge BX, as indicated in
the figure to the right. Then

a(p^2 +rs) =b^2 r+c^2 s.

Proof. We set θ = ABĈ ; applying the Law of Cosines to 4 AXB
yields

cosθ =

r^2 +p^2 −c^2

2 pr

.

Applying the Law of Cosines to the triangle 4 BXC gives