Two Perspective Triangles

Given ΔABC, the medial triangle M_aM_bM_c joins the midpoints of its sides, while the orthic triangle H_aH_bH_c joins the feet of its altitudes. Triangles ABC and M_aM_bM_c are homothetic in their common centroid, such that all of their remarkable points, lines and triangles are also homothetic. As a weak consequence, the corresponding objects of a triangle and its medial triangle are perspective. In particular, this is true for their orthic triangles.

Vladimir Nikolin observed (June, 2012) a simple, yet a non-trivial fact:

Proof

As the feet of two altitudes, H_a and H_b lie on the semicircle centered at M_c, implying that H_aM_c = H_bM_c. This makes ΔH_aM_cH_b isosceles so that the perpendicular bisector of the base H_aH_b passes through the vertex M_c. Since H_aH_b is a chord in the 9-point circle of ΔABC, the 9-point center N of ΔABC lies on M_cN_c. For the same reason, N lies on M_aN_a and M_bN_b; thus the three are indeed concurrent.

Nine Point Circle

• Nine Point Circle: an Elementary Proof
• Feuerbach's Theorem
• Feuerbach's Theorem: a Proof
• Four 9-Point Circles in a Quadrilateral
• Four Triangles, One Circle
• Hart Circle
• Incidence in Feuerbach's Theorem
• Six Point Circle
• Nine Point Circle
• 6 to 9 Point Circle
  • Six Concyclic Points
• Six Concyclic Points II
• Bevan's Point and Theorem
• Another Property of the 9-Point Circle
• Concurrence of Ten Nine-Point Circles
• Garcia-Feuerbach Collinearity
• Nine Point Center in Square

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