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XY bisects BC

Source: Thailand TSTST 2021, test 1, P3

August 16, 2022
geometryThailandcircumcircle

Problem Statement

An acute scalene triangle ABCABC with circumcircle Ω\Omega is given. The altitude from BB intersects side ACAC at B1B_1 and circle Ω\Omega at B2B_2. The circle with diameter B1B2B_1B_2 intersects circle Ω\Omega again at B3B_3. Similarly, the altitude from CC intersects side ABAB at C1C_1 and circle Ω\Omega at C2C_2. The circle with diameter C1C2C_1C_2 intersects circle Ω\Omega again at C3C_3. Let XX be the intersection of lines B1B3B_1B_3 and C1C3C_1C_3, and let YY be the intersection of lines B3CB_3C and C3BC_3B. Prove that line XYXY bisects side BCBC.
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