1966 AHSME Problems/Problem 31
Problem
Triangle is inscribed in a circle with center . A circle with center is inscribed in triangle . is drawn, and extended to intersect the larger circle in . Then we must have:
Solution
We will prove that and is isosceles, meaning that and hence .
Let and . Since the incentre of a triangle is the intersection of its angle bisectors, and . Hence . Since quadrilateral is cyclic, . So . This means that is isosceles, and hence .
Now let which means . Since is cyclic, Also, so . Thus, which means is isosceles, and hence .
Thus our answer is
See also
1966 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 30 |
Followed by Problem 32 | |
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