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Prove that two inscribed triangles are congruent

Source: Sharygin First Round 2013, Problem 15

April 7, 2013
trigonometrygeometrycircumcircleSharygin Geometry Olympiad

Problem Statement

(a) Triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 are inscribed into triangle ABCABC so that C1A1⊥BCC_1A_1 \perp BC, A1B1⊥CAA_1B_1 \perp CA, B1C1⊥ABB_1C_1 \perp AB, B2A2⊥BCB_2A_2 \perp BC, C2B2⊥CAC_2B_2 \perp CA, A2C2⊥ABA_2C_2 \perp AB. Prove that these triangles are equal.
(b) Points A1A_1, B1B_1, C1C_1, A2A_2, B2B_2, C2C_2 lie inside a triangle ABCABC so that A1A_1 is on segment AB1AB_1, B1B_1 is on segment BC1BC_1, C1C_1 is on segment CA1CA_1, A2A_2 is on segment AC2AC_2, B2B_2 is on segment BA2BA_2, C2C_2 is on segment CB2CB_2, and the angles BAA1BAA_1, CBB2CBB_2, ACC1ACC_1, CAA2CAA_2, ABB2ABB_2, BCC2BCC_2 are equal. Prove that the triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 are equal.
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