2003 AIME I Problems/Problem 15
In and Let be the midpoint of and let be the point on such that bisects angle Let be the point on such that Suppose that meets at The ratio can be written in the form where and are relatively prime positive integers. Find
In the following, let the name of a point represent the mass located there. Since we are looking for a ratio, we assume that , , and in order to simplify our computations.
First, reflect point over angle bisector to a point .
As is an angle bisector of both triangles and , we know that lies on . We can now balance triangle at point using mass points.
By the Angle Bisector Theorem, we can place mass points on of respectively. Thus, a mass of belongs at both and because BD is a median of triangle . Therefore, .
Now, we reassign mass points to determine . This setup involves and transversal . For simplicity, put masses of and at and respectively. To find the mass we should put at , we compute . Applying the Angle Bisector Theorem again and using the fact is a midpoint of , we find At this point we could find the mass at but it's unnecessary. and the answer is .
By the Angle Bisector Theorem, we know that . Therefore, by finding the area of triangle , we see that Solving for yields Furthermore, , so Now by the identity , we get But then , so . Thus .
Now by the Angle Bisector Theorem, , and we know that so .
We can now use mass points on triangle CBD. Assign a mass of to point . Then must have mass and must have mass . This gives a mass of . Therefore, , giving us an answer of
Let and . Then because is a median we have . Now we know Expressing the area of in two ways we have so Plugging this in we have so . But , so this simplifies to , and thus , and .
Solution 4 (Overpowered Projective Geometry!!)
Firstly, angle bisector theorem yields . We're given that . Therefore, the cross ratio
We need a fourth point for this cross ratio to be useful, so reflect point over angle bisector to a point . Then is isosceles and is an altitude so . Therefore,
All that's left is to fiddle around with the ratios:
Solution 5 (Menelaus + Mass Points)
Extend to intersect with the extension of at . Notice that , so . We now use Menelaus on , as , , and are collinear; this gives us . As , we have , hence . Reflect over to . Note that , and reflexivity, hence . It's easily concluded from this that , hence . As , we have is a midsegment of , thus . We now focus on the ratio . From similarity, we have . By the angle bisector theorem, we have , hence , so . We now work out the ratio . . We now use mass points on . We let the mass of be , so the mass of is and the mass of is . Hence, the mass of is , so the ratio . Extracting gives
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