2024 AMC 12B Problem 24

What is the number of ordered triples (a,b,c) of positive integers, with a\le b\le c\le 9, such that there exists a (non-degenerate) triangle \triangle ABC with an integer inradius for which a, b, and c are the lengths of the altitudes from A to \overline{BC}, B to \overline{AC}, and C to \overline{AB}, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)

\textbf{(A) }2\qquad \textbf{(B) }3\qquad \textbf{(C) }4\qquad \textbf{(D) }5\qquad \textbf{(E) }6\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.