This is really tough geometry problem?

Question

In triangle ABC, angle bisectors A,A1 and C,C1intersect at point E. Quadrilateral A1,B,C1,E are cyclic. Find the area of triangle AEC if AE=p and CE=q.

Answer 1

The answer is p² + pq + q²

To understand why, draw out the bisectors and the circle circumscribing the quadirilateral A1, B, C1, E. Let a and b be the angles A1AC and C1CA. Then a little work gives the tollowing values for angles:

ABC = 180 - 2(a+b)
A1EC1 = 180 - (a+b)

But since A1BC1E is cyclic, ABC + A1EC1 = 180, so we end up with a + b = 60, or AEC = 120. Using the cosine law for the area of the triangle AEC gets you the final result.

Answer 2

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