2004 Indonesia MO Problems/Problem 7
Contents
[hide]Problem
Prove that in a triangle with as the right angle, where denote the side in front of angle , denote the side in front of angle , denote the side in front of angle , the diameter of the incircle of equals to .
Solution
Let be the center of the circle, be the intersection of the incircle and , be the intersection of the incircle and , and be the intersection of the incircle and .
Note that and are tangent points of the circle, so and Since , we know that is a square, so
Let , , and Since are tangent points to the incircle, we know that and Thus,
Adding the three equations yields
Thus, so the diameter of the incircle is .
Solution 2
The inradius of any triangle has , where is the radius of the inscribed circle, is the semi-perimeter of the triangle, and is the area of the triangle. Thus,
If we multiply both sides by the "conjugate" of , we get
See Also
2004 Indonesia MO (Problems) | ||
Preceded by Problem 6 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 8 |
All Indonesia MO Problems and Solutions |