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Easy geometry [ISI (BS) 2006 #5]

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June 8, 2012
geometrytrigonometry

Problem Statement

Let A,BA,B and CC be three points on a circle of radius 11.
(a) Show that the area of the triangle ABCABC equals 12(sin⁡(2∠ABC)+sin⁡(2∠BCA)+sin⁡(2∠CAB))\frac12(\sin(2\angle ABC)+\sin(2\angle BCA)+\sin(2\angle CAB))
(b) Suppose that the magnitude of ∠ABC\angle ABC is fixed. Then show that the area of the triangle ABCABC is maximized when ∠BCA=∠CAB\angle BCA=\angle CAB
(c) Hence or otherwise, show that the area of the triangle ABCABC is maximum when the triangle is equilateral.
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