2018 AMC 12B Problems/Problem 21
Contents
Problem
In with side lengths , , and , let and denote the circumcenter and incenter, respectively. A circle with center is tangent to the legs and and to the circumcircle of . What is the area of ?
Diagram
~MRENTHUSIASM
Solution
We place the diagram in the coordinate plane: Let and
Since is a right triangle with its circumcenter is the midpoint of from which Let and denote the area and the semiperimeter of respectively. The inradius of is from which
Since is tangent to both coordinate axes, its center is at and its radius is for some positive number Let be the point of tangency of and As and are both perpendicular to the common tangent line at it follows that and are collinear.
Note that the circumradius of is We have or Solving this equation, we get
Finally, we find the area of by the Shoelace Theorem:
~pieater314159 ~MRENTHUSIASM
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 | |
All AMC 12 Problems and Solutions |
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