2011 AIME I Problems/Problem 4
In triangle , , and . The angle bisector of angle intersects at point , and the angle bisector of angle intersects at point . Let and be the feet of the perpendiculars from to and , respectively. Find .
Extend and such that they intersect line at points and , respectively. Since is the angle bisector of angle and is perpendicular to , must be an isoceles triangle, so , and is the midpoint of . For the same reason, , and is the midpoint of . Hence . Since so .
Let be the incenter of . Since lies on and , and , so . This means that is a cyclic quadrilateral. By the Law of Sines, , where is the radius of the circumcircle of . Since , we have that . Letting be the point of contact of the incircle of with side , we have . Thus, .
Solution 3 (Bash)
Project onto and as and . and are both in-radii of so we get right triangles with legs (the in-radius length) and . Since is the hypotenuse for the 4 triangles ( and ), are con-cyclic on a circle we shall denote as which is also the circumcircle of and . To find , we can use the Law of Cosines on where is the center of . Now, the circumradius can be found with Pythagorean Theorem with or : . To find , we can use the formula and by Heron's, . To find , we can find since . . Thus, and since , we have . Plugging this into our Law of Cosines (LoC) formula gives . To find , we use LoC on . Our formula now becomes . After simplifying, we get .
Because , is cyclic.
Applying Ptolemy's theorem on CMIN:
by sine angle addition formula.
Let be where the incircle touches , then . , for a final answer of .
Note: This is similar to Solution 2 after the first four lines
https://www.youtube.com/watch?v=yIUBhWiJ4Dk ~Mathematical Dexterity
Alternate Solution: https://www.youtube.com/watch?v=L2OzYI0OJsc&t=12s
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