Olympiad 3

Exercise 18. If the area of a regular hexagon ABCDEF is 54 √ 3 cm 2 and
AB = x cm, find x.
Exercise 19. The Angle bisector theorem.
Theorem 1. Consider triangle ABC.
Let the angle bisector of angle A intersect
side BC at a point D between
B and C. The angle bisector theorem
states that
BD
CD = AB
AC
Prove this theorem.
Exercise 20. Given triangle ABC such that AB = c, AC = b and BC = a,
(figure below).
A
A
• •
B
D
C
c
R
•
b
B
a
C
1. Show that a = c sin A
sin C , b = a sin B
b sin A
and c =
sin A
sin B . Hence, or
otherwise, show that the area of triangle ABC can be obtained as
A = a2 sin B sin C
2 sin A
= b2 sin A sin C
2 sin B
= c2 sin A sin B
2 sin C
2. Let R be a radius of circumscribed circle of △ABC, so that
R =
a
2 sin A = b
2 sin B = c
2 sin C
3. The Area of a triangle in terms of the radius of circumcircle
and angles, show that
A = 2R 2 sin A sin B sin C
4 Score: