1996 USAMO Problems/Problem 3
Let be a triangle. Prove that there is a line (in the plane of triangle ) such that the intersection of the interior of triangle and the interior of its reflection in has area more than the area of triangle .
Let the triangle be . Assume is the largest angle. Let be the altitude. Assume , so that . If , then reflect in . If is the reflection of , then and the intersection of the two triangles is just . But , so has more than the area of .
If , then reflect in the angle bisector of . The reflection of is a point on the segment and not . (It lies on the line because we are reflecting in the angle bisector. because . Finally, because we assumed does not exceed ). The intersection is just . But .
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