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Olympiad 3

Exercise 18. If the area

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:

The area of a triangle in terms of sides and the radius of the circumcircle, show that A = abc 4R 4. Let r be a radius of inscribed circle of △ABC and p = a + b + c is the 2 semi-perimeter. Show that A = rp. 5. Show that sin A √ (p − b)(p − c) 2 = bc you get tan A 2 = √ (p − b)(p − c) p(p − a) and cos A √ p(p − a) 2 = , hence bc 6. Using the figure below, show that x A = p−a, x B = p−b and x C = s−c, then show that tan A 2 = r p − a , tan B 2 = r p − b , tan C 2 = r p − c √ (p − a)(p − b)(p − c) Hence, show that r = p A x A x A B c r r b • x B x C r x B a x C C Therefore, we get the Heron’s formula A = √ p(p − a)(p − b)(p − c) Exercise 21. Given △ABC, show that: 1. sin A + sin B + sin C = 4 cos A 2 cos B 2 cos C 2 5 Score:

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