2007 AIME I Problems/Problem 15
Let be an equilateral triangle, and let and be points on sides and , respectively, with and . Point lies on side such that angle . The area of triangle is . The two possible values of the length of side are , where and are rational, and is an integer not divisible by the square of a prime. Find .
Denote the length of a side of the triangle , and of as . The area of the entire equilateral triangle is . Add up the areas of the triangles using the formula (notice that for the three outside triangles, ): . This simplifies to . Some terms will cancel out, leaving .
First of all, assume , then we can find It is not hard to find , we apply LOC on , getting that , leads to Apply LOC on separately, getting Add those terms together and use the equality , we can find:
According to basic angle chasing, , so , the ratio makes , getting that Now we have two equations with , and values for both equations must be the same, so we can solve for in two equations. , then we can just use positive sign to solve, simplifies to , getting , since the triangle is equilateral, , and the desired answer is
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