Question about a simple triangle property

General Questions
#1

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.

The GQ:

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?

#2

You just did it. Both give the area, and the area can’t change, so they must be the same.

#3

Similar triangles. Let ABC be the triangle, with the right angle at C. The triangles ABC and CBD share the angle at B, and so the ratios of their sides are the same. In particular, BC/CD = AC/AB, or in your language b/c = a/h.

#4

Psst, MikeS, that should be b/c = h/a.

#5

I think MikeS has the concept (if you want to omit the equal area proof), but with a typo.
In case it’s not clear, D is the point on the original hypotenuse (AB) which forms the height (right-angle line from C to AB). And A is the point opposite segment a, etc.
Then his last line should be
BC/CD=AB/AC, or in your language a/h = c/b, so ab = ch

(On preview, what Chronos said)

#6

That’ll teach me to dash off a quick post without carefully checking it. Yes, the point D is the intersection of the hypotenuse and its perpendicular line passing through C; and my equation should in fact read either b/c = h/a or c/b = a/h.

#7

½ab=area AND area= ½ch
So we have
½ab = ½ch
Multiply each side by 2 and you’re done.