Squares Inscribed In a Triangle III: What is this about?
A Mathematical Droodle

Explanation

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There are two ways to inscribe a square into a triangle, i.e. construct a square with vertices on the side lines of a given triangle. Since there are more points (vertices of the square) than lines (sides of the triangle), two of the vertices must lie on the same side. We may choose to place on a side two adjacent or two opposite vertices of the square. The former case has been discussed elsewhere. Here we consider the second possibility.

Assume two opposite vertices of a square are located on the side line AB of triangle ABC. Consider the orthic triangle H_aH_bH_c. The construction is based on a property of point L - the intersection of AB and H_aH_b: the perpendicular to AB at L meets the remaining side lines AC and BC at the points L_b and L_a equidistant from L. Based on this property, the sought square has L as the center and L_b and L_a as two of the vertices. The remaining vertices that lie on AB can be now found quite easily.

If we sought two opposite vertices of the square on sides BC or AC, instead of AB we would have two additional points M and N as their centers. As a matter of fact, the three points L, M, and N are collinear. The line they lie on is known as the orthic axis of triangle ABC.

References

1. F. van Lamoen, Inscribed Squares, Forum Geometricorum, Volume 4 (2004) 207-214

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