Begin with a right triangle with hypotenuse
h.
Inscribe a circle and label its radius, r.
What is the ratio of the area of the circle to that of the triangle?
I had the pleasure of drawing the figure and then having it instruct me how to simply write down the answer: Naturally, inscribe the circle in the triangle, and draw in the three radii that are perpendicular to the sides. Lastly, draw lines from the circle center to the two vertices other than the right angle. You will see that the triangle has been divided into five shapes: two pairs of congruent right triangles and a square of side r. The hypotenuse of the original triangle is now divided into lengths h-x and x. There are two right triangles with (base, height) = (h-x, r) and two with (base, height) = (x, r). Adding these four areas gives (r h). Including the square gives r h+r^2 for the triangle's full area. So the ratio is pi r^2 / (r h + r^2) = pi r/(r+h). QED
(The given solution is very simple too, but it says "We see that h=x+y-2r", which I believe, but I really don't see right off...)
Edited on April 11, 2021, 6:56 pm
simpler yet
