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Find the smallest value of $\frac{AO_1+BO_1}{AB},$

Source: Moldova TST 2016

January 30, 2023
geometrycircumcircle

Problem Statement

Let ABCABC be a triangle with ∠C=90\angle C=90. The tangent points of the inscribed circle with the sides BC,CABC, CA and ABAB are M,NM, N and P.P. Points M1,N1,P1M_1, N_1, P_1 are symmetric to points M,N,PM, N, P with respect to midpoints of sides BC,CABC, CA and AB.AB. Find the smallest value of AO1+BO1AB,\frac{AO_1+BO_1}{AB}, where O1O_1 is the circumcenter of triangle M1N1P1.M_1N_1P_1.
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