2019 AIME II Problems/Problem 15
In acute triangle points and are the feet of the perpendiculars from to and from to , respectively. Line intersects the circumcircle of in two distinct points, and . Suppose , , and . The value of can be written in the form where and are positive integers, and is not divisible by the square of any prime. Find .
First we have , and Similarly, and dividing these each by gives .
It is known that the sides of the orthic triangle are , and its angles are ,, and . We thus have the three sides of the orthic triangle now. Letting be the foot of the altitude from , we have, in , similarly, we get Our final answer is then
The requested sum is .
By power of point, we have Which are simplified to
In triangle , by law of cosine
Substitute everything by
The quadratic term is cancelled out after simplified
Plug back in,
So the final answer is
By power of point, we have and
Therefore, substituting in the values:
Notice that quadrilateral is cyclic.
From this fact, we can deduce that and
Therefore is similar to .
Now using Law of Cosines on we get:
Substituting and Simplifying:
Now we solve for using regular algebra which actually turns out to be very easy.
We get and from the above relations between the variables we quickly determine , and
So the answer is
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