Given a triangle Δ A B C, an inscribed square is a square all four of whose vertices lie on the edges of Δ A B C and two of whose vertices fall on the same edge. As noted by van Lamoen, there are two types of squares inscribing reference triangle Δ A B C in the sense that all vertices lie on the sidelines of A B C. In particular, the first type has two adjacent vertices of the square on one side, while the second type has two opposite vertices on one side. There are three squares of each type, and van Lamoen gives centers and vertices of the three squares of each type in homogeneous barycentric coordinates.

Ehrmann congruent squares point | Lucas circles | square | square inscribing | triangle