On the plane there are two triangles, each with side lengths 18, 24 and 30. If the two triangles do not completely overlap, but share the same circumcircle as well as the same inscribed circle, find the area of the region common to both triangles.

Note that 18^2+24^2=30^2.  Actually its a 3-4-5 right triangle scaled by 6.

For two right triangles to have congruent circumcircles and congruent incircles, they must be congruent.  To have the very same circumcircle and incircle they must be reflection images over the line containing the centers of these circles.

Construct the right triangle, its circumcenter and its incenter.

Construct the line though these centers.
Reflect the triangle over these lines.
The intersection of the triangle and its reflection is a hexagon.
The area of this hexagon is 132.

Posted by Jer on 2014-02-22 11:50:18