In traingle ABC, point X is on side BC such that AX=13, BX=10, and CX=4, and the circumcircles of triangles ABX and ACX have the same radius. Find the area of triangle ABC.
Let r be the radius of the circumcircles of triangles ABX and ACX. Then the area of triangle ABX is 21(AB)(r)=21(13)(r), and the area of triangle ACX is 21(AC)(r)=21(14)(r). Since the circumcircles have the same radius, the areas of the triangles are equal. Therefore, 21(13)(r)=21(14)(r), or r=13.
The area of triangle ABC is 1/2*(AB)(AC)=1/2*(13)(14)=91.