Difference between revisions of "2005 AIME I Problems/Problem 15"
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== Solution 2 == | == Solution 2 == | ||
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WLOG let E be be between C & D (as in solution 1). Assume <math>AD = 3m</math>. We use power of a point to get that | WLOG let E be be between C & D (as in solution 1). Assume <math>AD = 3m</math>. We use power of a point to get that | ||
<math>AG = DE = \sqrt{2}m </math> and <math>AB = AG + GB = AG + BE = 10+2\sqrt{2} m</math> | <math>AG = DE = \sqrt{2}m </math> and <math>AB = AG + GB = AG + BE = 10+2\sqrt{2} m</math> |
Revision as of 22:53, 18 January 2021
Contents
Problem
Triangle has The incircle of the triangle evenly trisects the median If the area of the triangle is where and are integers and is not divisible by the square of a prime, find
Solution 1
Let , and be the points of tangency of the incircle with , and , respectively. Without loss of generality, let , so that is between and . Let the length of the median be . Then by two applications of the Power of a Point Theorem, , so . Now, and are two tangents to a circle from the same point, so by the Two Tangent Theorem and thus . Then so and thus .
Now, by Stewart's Theorem in triangle with cevian , we have
Our earlier result from Power of a Point was that , so we combine these two results to solve for and we get
Thus or . We discard the value as extraneous (it gives us a line) and are left with , so our triangle has area and so the answer is .
Solution 2
WLOG let E be be between C & D (as in solution 1). Assume . We use power of a point to get that and
Since now we have , in triangle and cevian . Now, we can apply Stewart's Theorem.
or if , we get a degenerate triangle, so , and thus . You can now use Heron's Formula to finish. The answer is , or .
-Alexlikemath
See also
2005 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.