TITLE: A geometric look at $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$?
QUESTION [18 upvotes]: Is there a geometric way of looking at the relationship between the positive real numbers $a$, $b$ and $c$ if $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$?

REPLY [2 votes]: In a right-angled triangle (with legs a and b, and hypotenuse c) you can inscribe a square in two "obvious" ways:

The square shares the right angle with the triangle and touches c. Then 1/s = 1/a + 1/b
One side of the square lies on c, and the square touches a and b. Then 1/t = 1/c + 1/h (where h is the height on c)