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concurrent wanted, perimeters equal to CA+AB,AB +BC,BC +CA

Source: 2018 Thailand October Camp 2.3

October 15, 2020
geometryconcurrencyconcurrentparallelperimeter

Problem Statement

Let ABCABC be an acute triangle with AX,BYAX, BY and CZCZ as its altitudes. ∙\bullet Line ℓA\ell_A, which is parallel to YZYZ, intersects CACA at A1A_1 between CC and AA, and intersects ABAB at A2A_2 between AA and BB. ∙\bullet Line ℓB\ell_B, which is parallel to ZXZX, intersects ABAB at B1B_1 between AA and BB, and intersects BCBC at B2B_2 between BB and CC. ∙\bullet Line ℓC\ell_C, which is parallel to XYXY , intersects BCBC at C1C_1 between BB and CC, and intersects CACA at C2C_2 between CC and AA. Suppose that the perimeters of the triangles △AA1A2\vartriangle AA_1A_2, △BB1B2\vartriangle BB_1B_2 and △CC1C2\vartriangle CC_1C_2 are equal to CA+AB,AB+BCCA+AB,AB +BC and BC+CABC +CA, respectively. Prove that ℓA,ℓB\ell_A, \ell_B and ℓC\ell_C are concurrent.
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