what's the name of the theorem:median of right-triangle hypotenuse is always half of it

Solution 1:

Here is a proof without words:

Solution 2:

If you are interested in seeing proofs: see these proofs. You'll actually find two proofs (and illustrations) of the following theorems:

1. If a triangle is a right triangle, the median drawn to the hypotenuse has the measure half the hypotenuse (from which it follows that the median drawn to the hypotenuse divides the triangle in two isosceles triangles); and

2. If in a triangle a median has the measure half the length of the side it is drawn, then the triangle is a right triangle.

The proofs are not at all elaborate, and utilize properties you already know, so they are easy enough to reconstruct, if necessity dictates that you do so. Understanding "why" these theorems hold is the important point; correctly "naming" them is a less important.