1999 AIME Problems/Problem 12
Problem
The inscribed circle of triangle is tangent to at and its radius is . Given that and find the perimeter of the triangle.
Solution
Solution 1
Let be the tangency point on , and on . By the Two Tangent Theorem, , , and . Using , where , we get . By Heron's formula, . Equating and squaring both sides,
We want the perimeter, which is .
Solution 2
Let the incenter be denoted . It is commonly known that the incenter is the intersection of the angle bisectors of a triangle. So let and
We have that So naturally we look at But since we have Doing the algebra, we get
The perimeter is therefore
solution 3
Let unknown side has length as , Assume three sides of triangles are , the area of the triangle is .
Note that
. Use trig identity, knowing that , getting that
Now equation , the final answer is ~bluesoul
See also
1999 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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.