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AA_1, BB_1 , CC_1 are concurrent iff < BAC = 90^o

Source: Moldova JBMO TST 2000 2.3

July 9, 2020
geometryconcurrencyconcurrentright angle

Problem Statement

Let a triangle ABC,A1ABC, A_1 be the midpoint of the segment [BC],B1∈(AC)[BC], B_1 \in (AC) ¸and C1∈(AB)C_1 \in (AB) such that [A1B1[A_1B_1 is the bisector of the angle AA1CAA_1C and A1C1A_1C_1 is perpendicular to ABAB. Show that the lines AA1,BB1AA_1, BB_1 and CC1CC_1 are concurrent if and only if ∠BAC=90o \angle BAC = 90^o
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