1988 IMO Problems/Problem 5
Problem
In a right-angled triangle let be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles intersect the sides at the points respectively. If and dnote the areas of triangles and respectively, show that
Solution
Lemma: Through the incenter of draw a line that meets the sides and at and , then: Proof of the lemma: Consider the general case: is any point on side and is a line cutting AB, AM, AC at P, N, Q. Then:
If is the incentre then , and . Plug them in we get:
Back to the problem Let and be the areas of and and be the intersection of and . Thus apply our formula in the two triangles we get: and Cancel out the term , we get: So we conclude .
Hence and , thus and . Thus . So the area ratio is:
This solution was posted and copyrighted by shobber. The original thread for this problem can be found here: [1]
See Also
1988 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
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