1993 AIME Problems/Problem 15
Problem
Let be an altitude of . Let and be the points where the circles inscribed in the triangles and are tangent to . If , , and , then can be expressed as , where and are relatively prime integers. Find .
Solution
From the Pythagorean Theorem, , and .
Subtracting those two equations yields .
After simplification, we see that , or .
Note that .
Therefore we have that .
Therefore .
Now note that , , and .
Therefore we have .
Plugging in and simplifying, we have .
Edit by GameMaster402:
It can be shown that in any triangle with side lengths , if you draw an altitude from the vertex to the side of , and draw the incircles of the two right triangles, the distance between the two tangency points is simply .
Plugging in yields that the answer is , which simplifies to
~minor edit by Yiyj1
Edit by phoenixfire:
It can further be shown for any triangle with sides that Over here , so using the formula gives
~minor edit by Yiyj1
Note: We can also just right it as since by the triangle inequality. ~Yiyj1
See also
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