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Problem 5

Part of 2021 Junior Macedonian Mathematical Olympiad

Problems(1)

Ratio of distances from incenters

Source: 2021 Junior Macedonian Mathematical Olympiad P5

6/8/2021
Let ABCABC be an acute triangle and let XX and YY be points on the segments ABAB and ACAC such that BX=CYBX = CY. If IBI_{B} and ICI_{C} are centers of inscribed circles in triangles ABYABY and ACXACX, and TT is the second intersection point of the circumcircles of ABYABY and ACXACX, show that:
TIBTIC=BYCX.\frac{TI_{B}}{TI_{C}} = \frac{BY}{CX}.
Proposed by Nikola Velov
ratiogeometryincenter