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 1 of 4: Translate the median condition

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In plain words

A right triangle is inscribed in the circle with the hypotenuse as diameter, so its hypotenuse median is the circle's radius.

m=\frac{c}{2},\qquad ab=m^2=\frac{c^2}{4}

Detailed analysis

Let the legs be $a,b$ . The median to the hypotenuse has length $m=c/2$ . Requiring it to be the geometric mean of the legs gives $m=\sqrt{ab}$ , hence $ab=c^2/4$ .