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side ratio in triangle, angle bisectors form cyclic quadrilateral

Source: Mongolia MO 2000 Grade 10 P6

April 22, 2021
geometryangle bisectorratiocyclic quadrilateral

Problem Statement

In a triangle ABCABC, the angle bisector at A,B,CA,B,C meet the opposite sides at A1,B1,C1A_1,B_1,C_1, respectively. Prove that if the quadrilateral BA1B1C1BA_1B_1C_1 is cyclic, then ACAB+BC=ABAC+CB+BCBA+AC.\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.
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