midpoint ellipse

The Steiner inellipse, also called the midpoint ellipse, is an inellipse with inconic parameters x:y:z = a:b:c giving equation a^2 α^2 + b^2 β^2 + c^2 γ^2 - 2a b αβ - 2a c αγ - 2b c βγ = 0. It has the triangle centroid G as both its center and Brianchon point. It is tangent to the sides of the triangle at their midpoints, so the triangle formed by its contact points with the reference triangle is the medial triangle, which is also its polar triangle. This conic is always an ellipse.

affine transformation | Brianchon point | inconic | inellipse | medial triangle | midpoint | Steiner circumellipse

Jakob Steiner