Advanced High-School Mathematics

SECTION 1.2 Triangle Geometry 25

Proof. Referring to the dia-
gram to the right and using the
Pythagorean Theorem, we infer
quickly that

c^2 = (b−acosC)^2 +a^2 sin^2 C

= b^2 − 2 abcosC+a^2 cos^2 C+a^2 sin^2 C

= a^2 +b^2 − 2 abcosC,

as required.

Exercises

1. Using the Law of Sines, prove the Angle Bisector Theorem (see
   page 15).


2. Prove Heron’s formula. Namely, for the triangle whose side
   lengths area, b,andc, prove that the area is given by

area =

√

s(s−a)(s−b)(s−c),

where s =

a+b+c

2

= one-half the perimeter of the triangle.

(Hint: ifAis the area, then start with 16A^2 = 4b^2 (c^2 −c^2 cos^2 A) =

(2bc− 2 bccosA)(2bc+ 2bccosA). Now use the Law of Cosines to

write 2bccosAin terms ofa, b,andcand do a bit more algebra.)

3. In the quadrilateral depicted at the
   right, the lengths of the diagonals
   areaandb, and meet at an angleθ.
   Show that the area of this quadri-
   lateral is^12 absinθ. (Hint: compute
   the area of each triangle, using the
   Law of Sines.)

a

b

θ