1999 AIME Problems/Problem 12
The inscribed circle of triangle is tangent to at and its radius is . Given that and find the perimeter of the triangle.
Let be the tangency point on , and on . By the Two Tangent Theorem, , , and . Using , where , we get . By Heron's formula, . Equating and squaring both sides,
We want the perimeter, which is .
Let the incenter be denoted . It is commonly known that the incenter is the intersection of the angle bisectors of a triangle. So let and
We have that So naturally we look at But since we have Doing the algebra, we get
The perimeter is therefore
Let unknown side has length as , Assume three sides of triangles are , the area of the triangle is .
. Use trig identity, knowing that , getting that
Now equation , the final answer is ~bluesoul
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