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How on earth do you write problems like these?

Source: Sharygin Correspondence Round 2024 P20

March 6, 2024
geometryisogonalscircumcircle

Problem Statement

Lines a1,b1,c1a_1, b_1, c_1 pass through the vertices A,B,CA, B, C respectively of a triange ABCABC; a2,b2,c2a_2, b_2, c_2 are the reflections of a1,b1,c1a_1, b_1, c_1 about the corresponding bisectors of ABCABC; A1=b1∩c1,B1=a1∩c1,C1=a1∩b1A_1 = b_1 \cap c_1, B_1 = a_1 \cap c_1, C_1 = a_1 \cap b_1, and A2,B2,C2A_2, B_2, C_2 are defined similarly. Prove that the triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 have the same ratios of the area and circumradius (i.e. S1R1=S2R2\frac{S_1}{R_1} = \frac{S_2}{R_2}, where Si=S(△AiBiCi)S_i = S(\triangle A_iB_iC_i), Ri=R(△AiBiCi)R_i = R(\triangle A_iB_iC_i))
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