MathDB
Special line through antipodal

Source: 2025 Israel TST Test 1 P2

October 28, 2024
geometryconfigurationscircumcircleincenter

Problem Statement

Triangle △ABC\triangle ABC is inscribed in circle Ω\Omega. Let II denote its incenter and IAI_A its AA-excenter. Let NN denote the midpoint of arc BACBAC. Line NIANI_A meets Ω\Omega a second time at TT. The perpendicular to AIAI at II meets sides ACAC and ABAB at EE and FF respectively. The circumcircle of △BFT\triangle BFT meets BIABI_A a second time at PP, and the circumcircle of △CET\triangle CET meets CIACI_A a second time at QQ. Prove that PQPQ passes through the antipodal to AA on Ω\Omega.
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