Can you correct Calvin's mistake?

Geometry Level 5

Calvin was giving a class on triangles, and he was planning to demonstrate on the blackboard that the three medians, the three angle bisectors, and the three altitudes of a triangle each meet at a point (the centroid, incentre, and orthocentre of the triangle, respectively).

Unfortunately, he got a little careless in his example, and drew a certain triangle \(ABC\) with the median from vertex \(A\), the altitude from vertex \(B\), and the angle bisector from vertex \(C\). Amazingly, just as he discovered his mistake, he saw that the three segments met at a point anyway!

Luckily it was the end of the period, so no one had a chance to comment on his mistake. In recalling his good fortune later that day, he could only remember that the side across from vertex \(C\) was \(13\) inches in length, that the other two sides also measured an integral number of inches, and that none of the lengths were the same. Can you help Calvin in finding the sum of the other two lengths?

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