And, no, it definitely ain’t homework.
Let’s say I have a garden variety right triangle with sides a, b, and c, with c being the hypotenuse.
Currently, it’s sitting flat on one if it sides with the right angle on the left, and the hypotenuse sloping downward to the right.
I take the same triangle, and set it such that it is lying on its hypotenuse. It has a height h formed by dropping a line from the angle between a and b to the hypotenuse.
The area can be determined by either ½ab or ½ch. So, the product of the two short sides equals the hypotenuse times the height.
What’s the easiest way of showing that in a right triangle that ab = ch? I know you can grind it out formulaicly, but what’s the slickest way?