2006 AMC 10B Problems/Problem 24
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Problem
Circles with centers 
and 
have radii 
and 
, respectively, and are externally tangent. Points 
and 
on the circle with center and points 
and 
on the circle with center are such that 
and 
are common external tangents to the circles. What is the area of the concave hexagon 
?
Solution
Since a tangent line is perpendicular to the radius containing the point of tangency, 
.
Construct a perpendicular to 
that goes through point . Label the point of intersection 
.
Clearly 
is a rectangle.
Therefore 
and 
.
By the Pythagorean Theorem:
.
The area of is 
.
The area of 
is 
.
So the area of quadrilateral 
is 
.
Using similar steps, the area of quadrilateral 
is also 
Therefore, the area of hexagon is 
