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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.

 Oct 28, 2023
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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​.

 Oct 28, 2023

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