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Romania TST 2022 Day 4 P3

Source: Romania TST 2022

June 3, 2022
geometryromaniaRomanian TST

Problem Statement

Let ABCABC be a triangle and let its incircle γ\gamma touch the sides BC,CA,ABBC,CA,AB at D,E,FD,E,F respectively. Let PP be a point strictly in the interior of γ.\gamma. The segments PA,PB,PCPA,PB,PC cross γ\gamma at A0,B0,C0A_0,B_0,C_0 respectively. Let SA,SB,SCS_A,S_B,S_C be the centres of the circles PEF,PFD,PDEPEF,PFD,PDE respectively and let TA,TB,TCT_A,T_B,T_C be the centres of the circles PB0C0,PC0A0,PA0B0PB_0C_0,PC_0A_0,PA_0B_0 respectively. Prove that SATA,SBTBS_AT_A, S_BT_B and SCTCS_CT_C are concurrent.
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