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12

Part of 2023 Sharygin Geometry Olympiad

Problems(1)

perpendicular bisectors featuring circles

Source: Sharygin 2023 - P12 (Grade-(a)8-9, (b)10-11)

3/4/2023
Let ABCABC be a triangle with obtuse angle BB, and P,QP, Q lie on ACAC in such a way that AP=PB,BQ=QCAP = PB, BQ = QC. The circle BPQBPQ meets the sides ABAB and BCBC at points NN and MM respectively.
<spanclass=′latex−bold′>(a)</span>\qquad<span class='latex-bold'>(a)</span> (grades 8-9) Prove that the distances from the common point RR of PMPM and NQNQ to AA and CC are equal. <spanclass=′latex−bold′>(b)</span>\qquad<span class='latex-bold'>(b)</span> (grades 10-11) Let BRBR meet ACAC at point SS. Prove that MN⊥OSMN \perp OS, where OO is the circumcenter of ABCABC.
geometryperpendicular bisectororthologic trianglesSharygin Geometry OlympiadSharygin 2023