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circles related to orthoc triangle

Source: Eotvos 1909 p3

September 8, 2024
geometryorthic triangle

Problem Statement

Let A1,B1,C1A_1, B_1, C_1, be the feet of the altitudes of △ABC\vartriangle ABC drawn from the vertices A,B,CA, B, C respectively, and let MM be the orthocenter (point of intersection of altitudes) of △ABC\vartriangle ABC. Assume that the orthic triangle (i.e. the triangle whose vertices are the feet of the altitudes of the original triangle) A1A_1,B1B_1,C1C_1 exists. Prove that each of the points MM, AA, BB, and CC is the center of a circle tangent to all three sides (extended if necessary) of △A1B1C1\vartriangle A_1B_1C_1. What is the difference in the behavior of acute and obtuse triangles ABCABC?
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