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perpendicular wanted, starting with equilateral (2014 Kyiv City MO 8.5)

Source:

August 5, 2020
geometryperpendicularEquilateral

Problem Statement

Given an equilateral ΔABC\Delta ABC, in which A1,B1,C1{{A} _ {1}}, {{B} _ {1}}, {{C} _ {1}} are the midpoint of the sides BC,  AC,  AB BC, \, \, AC, \, \, AB respectively. The line ll passes through the vertex AA, we denote by P,QP, Q the projection of the points B,CB, C on the line ll, respectively (the line l l and the point Q,  A,  PQ, \, \, A, \, \, P are located as shown in fig.). Denote by TT the intersection point of the lines B1P{{B} _ {1}} P and C1Q{{C} _ {1}} Q. Prove that the line A1T{{A} _ {1}} T is perpendicular to the line ll. https://cdn.artofproblemsolving.com/attachments/4/b/61f2f4ec9e6b290dfcd47e9351110bebd3bd43.png (Serdyuk Nazar)
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