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

Question

This question is related to one of my previous questions.

The answer to that question included a theorem: "The median on the hypotenuse of a right triangle equals one-half the hypotenuse".

When I wrote the answer out and showed it a friend of mine, he basically asked me how I knew that the theorem was true, and if the theorem had a name.

So, my question:

-Does this theorem have a name?
-If not, what would be the best way to describe it during a math test? Or is it better to write out the full prove every time?

Answer 1

You could call it a special case of Apollonius's theorem, or of the parallelogram law.

Answer 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.

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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.

Answer 3

In a circle, a triangle which has for side a diameter is rectangle. This means the center of the circle is the mean point of the diameter, and the median is a radius of the circle.

You can prove this as a corollary to the fact that inscribed angles are half of the center ones. So I guess that is a corollary to this corollary.

Answer 4

Here is a proof without words: