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Geometry Mathley 5.3 3 lines concurrent with Euler line

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June 7, 2020
Euler Lineconcurrentgeometry

Problem Statement

Let ABCABC be an acute triangle, not being isoceles. Let ℓa\ell_a be the line passing through the points of tangency of the escribed circles in the angle AA with the lines AB,ACAB, AC produced. Let dad_a be the line through AA parallel to the line that joins the incenter II of the triangle ABCABC and the midpoint of BCBC. Lines ℓb,db,ℓc,dc\ell_b, d_b, \ell_c, d_c are defined in the same manner. Three lines ℓa,ℓb,ℓc\ell_a, \ell_b, \ell_c intersect each other and these intersections make a triangle called MNPMNP. Prove that the lines da,dbd_a, d_b and dcd_c are concurrent and their point of concurrency lies on the Euler line of the triangle MNPMNP.
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