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The unique Ellipse tangent to the Midpoints of a Triangle's Legs. The midpoint ellipse has the maximum Area of any Inscribed Ellipse (Chakerian 1979). Under an Affine Transformation, the midpoint ellipse can be transformed into the Incircle of an Equilateral Triangle.
See also Affine Transformation, Ellipse, Incircle, Midpoint, Triangle
References
Central Similarities. University of Minnesota College Geometry Project. Distributed by International Film Bureau, Inc.
Chakerian, G. D. ``A Distorted View of Geometry.'' Ch. 7 in Mathematical Plums (Ed. R. Honsberger).
Washington, DC: Math. Assoc. Amer., pp. 135-136 and 145-146, 1979.
Pedoe, D. ``Thinking Geometrically.'' Amer. Math. Monthly 77, 711-721, 1970.