George Rebane

Research has shown that if you spend some time noodling on problems like math/logic puzzles or play chess or other similar pursuits that stretch your brainbone, then that can help delay or even eliminate the onset of dementias.  As a public service I am doing my part in offering readers the prophylactic shown in the figure.

There we see a circle nested in the corner of a square and a line drawn from the opposite corner that is tangent to the circle.  This forms the shown blue right triangle.  Given that the diameter of the circle is a known fraction f (ranging from zero to one) of the length L of the square’s side, derive the formula for the area A of the triangle in terms of f and L.  A little thought reveals that A = 0 when f = 1, and A = L^2/2 when f = 0.  Use this to check your formula.