A right triangle has fixed base and variable height. An arc, centered at the peak of the triangle, passes through the right angle to the hypotenuse. The blue square is drawn so that it touches the arc and hypotenuse. It is the largest square that fits inside the triangle and under the arc. This is Problem 47 on page 120 of Sacred Mathematiocs: Japanese Temple Geometry, by Fukagawa Hidetoshi and Tony Rothman, where the focus is on maximizing the size of the blue square.
Drag the peak of the triangle up and down, and notice how the size of the blue square changes. Try to make the square as large as possible. What do you notice about where that happens?