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square inscribed in triangle, sum of reciprocals of 3 inradii

Source: 2020 Dürer Math Competition Finals Day2 E13 https://artofproblemsolving.com/community/c1622639_2020_

January 7, 2022
squaregeometryinscribedinradius

Problem Statement

In triangle ABCABC we inscribe a square such that one of the sides of the square lies on the side ACAC, and the other two vertices lie on sides ABAB and BCBC. Furthermore we know that AC=5AC = 5, BC=4BC = 4 and AB=3AB = 3. This square cuts out three smaller triangles from △ABC\vartriangle ABC. Express the sum of reciprocals of the inradii of these three small triangles as a fraction p/qp/q in lowest terms (i.e. with pp and qq coprime). What is p+qp + q?
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