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Intersections On The Euler Line

Source: 2020 RMM Shortlist G2

October 8, 2022
geometryEuler LineRMMRMM 2020RMM Shortlist

Problem Statement

Let ABCABC be an acute scalene triangle, and let A1,B1,C1A_1, B_1, C_1 be the feet of the altitudes from A,B,CA, B, C. Let A2A_2 be the intersection of the tangents to the circle ABCABC at B,CB, C and define B2,C2B_2, C_2 similarly. Let A2A1A_2A_1 intersect the circle A2B2C2A_2B_2C_2 again at A3A_3 and define B3,C3B_3, C_3 similarly. Show that the circles AA1A3,BB1B3AA_1A_3, BB_1B_3, and CC1C3CC_1C_3 all have two common points, X1X_1 and X2X_2 which both lie on the Euler line of the triangle ABCABC.
United Kingdom, Joe Benton
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