Problem 4

Construct a right triangle with a given hypotenuse

c

such that the median drawn to the hypotenuse is the geometric mean of the two legs.

Step 4 of 4: Verify necessity and sufficiency

Previous step Next step

\text{The construction is valid, and every valid triangle arises this way}

Detailed analysis

The preceding area argument proves every required triangle has altitude $c/4$ , so it must occur among these intersections. Conversely, every intersection gives a right triangle with hypotenuse $c$ and altitude $c/4$ , hence $ab=ch=c^2/4$ and $\sqrt{ab}=c/2$ , exactly the median length.