2016 AMC 10A Problems/Problem 21
Circles with centers and , having radii and , respectively, lie on the same side of line and are tangent to at and , respectively, with between and . The circle with center is externally tangent to each of the other two circles. What is the area of triangle ?
Notice that we can find in two different ways: and , so
. Additionally, . Therefore, . Similarly, . We can calculate easily because . .
Plugging into first equation, the two sums of areas, .
Use the Shoelace Theorem.
Let the center of the first circle of radius 1 be at (0, 1).
Draw the trapezoid and using the Pythagorean Theorem, we get that so the center of the second circle of radius 2 is at .
Draw the trapezoid and using the Pythagorean Theorem, we get that so the center of the third circle of radius 3 is at .
Now, we may use the Shoelace Theorem!
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