# Two Perspective Triangles

Given ΔABC, the medial triangle MaMbMc joins the midpoints of its sides, while the orthic triangle HaHbHc joins the feet of its altitudes. Triangles ABC and MaMbMc 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:

The medial triangles of a triangle and of its orthic triangle are perspective. In other words, lines MaNa, MbNb, McNc, where Na, Nb, and Nc are the midpoints of the orthic triangle HaHbHc, are concurrent. Furthermore, they are concurrent in the 9-point center of ΔABC.

### Proof

As the feet of two altitudes, Ha and Hb lie on the semicircle centered at Mc, implying that HaMc = HbMc. This makes ΔHaMcHb isosceles so that the perpendicular bisector of the base HaHb passes through the vertex Mc. Since HaHb is a chord in the 9-point circle of ΔABC, the 9-point center N of ΔABC lies on McNc. For the same reason, N lies on MaNa and MbNb; thus the three are indeed concurrent. ### Nine Point Circle 