Problem:
Isosceles triangle has , and a circle with radius is tangent to line at and to line at . What is the area of the circle that passes through vertices , and ?
Answer Choices:
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Solution:
Let be the center of the circle with radius . Consider the circle with diameter . Because and are right angles, the opposite angles of quadrilateral are supplementary, and hence this quadrilateral is cyclic. Thus is also on the circle that passes through , and , and by symmetry is a diameter. By the Pythagorean Theorem,
so the circle that passes through , and has radius and area .
The problems on this page are the property of the MAA's American Mathematics Competitions