Take any triangle ABC.
Develop a ruler and compass construction for inscribing a square in a triangle with one side along a specified base.
Help? See Squares Inscribed in a Right Triangle.
Given the inscribed square as indicated in the drawing, find an expression for s in terms of h and a.
This result shows that the length of the side of a square inscribed on a given base of a triangle is one half the harmonic mean of the altitude to that base and the base.
Another way to set it up.