2011 AMC 12A Problems/Problem 17
Circles with radii , , and are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
The centers of these circles form a 3-4-5 triangle, which has an area equal to 6.
The areas of the three triangles determined by the center and the two points of tangency of each circle are, using Triangle Area by Sine,
which add up to . The area we're looking for is the large 3-4-5 triangle minus the three smaller triangles, or .
Solution 2 (Analytical)
Let be the centers of the circles with radii . Notice that the points of tangency of the circles are also the points of tangency of the incircle of . Using the radius of an Incircle formula, where is the semi-perimeter, and noting that is a 3-4-5 right triangle, we see that, Now we set on the coordinate plane with . So the incenter lies on . Let the points of tangency of with it's incenter are with on , on , and on . We have that . Since the line defined by and the incenter is perpendicular to who has equation , we have it's equation as . We have the intersection of the lines at, Here we can use Shoelace Theorem on the points we get our areas as
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