2008 iTest Problems/Problem 96
Contents
Problem
Triangle has , and , and a point is chosen inside the triangle. The interior angle bisectors , and of respective angles , and intersect pairwise at , and . If triangles and are directly similar, then the area of may be written in the form , where are positive integers, and are not divisible by the square of any prime, and . Compute .
Solution
Let . With some angle chasing, we find that , , , and .
By using 30-60-90 triangles, we find that and . By the Law of Sines on and , and . After solving for in both equations, we have
Thus, by using identities, . Now we will determine the length of . We will only substitute at the very end (along with the other trigonometric expressions) to keep calculations simple.
Using the definition of a cosine, we have . By the Law of Sines on , , so . Thus .
We know that the area of is and that Therefore,
Therefore, .
Note
The original problem says that , and . This is a typo.
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 95 |
Followed by: Problem 97 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 • 100 |