AMC 12B 2024 (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

\overline{BC}

B

\overline{AC}

, and

C

\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.)

\text{(A)}\;2

\text{(B)}\;3

\text{(C)}\;4

\text{(D)}\;5

\text{(E)}\;6