$$PUTNAM,1986, QUESTION:B1$$ Inscribe a rectangle of base $b$ and height $h$ and an isosceles triangle of base $b$ in a circle of radius 1. For what value of $h$ do the rectangle and triangle have the same area.
I think it is a simple problem(in context of others), and I have solved it but I am looking for alternate solutions.
My solution: Make a similar isosceles triangle on the other side $b$ of the rectangle. If we see the line joining the apex of each isosceles triangle,it is divided in the ratio $ 2:1:2 $ by the side $b$.
This line segment is also the diameter and hence we get $h=\frac25$