A Suggestion

Take a segment of length a + b as the diameter of a semicircle. Construct a perpendicular from the common end of the segments of length a and length b and locate its intersection with the semicircle. Draw the segment from the midpoint of the diameter to that intersection.

The blue segment is always as long or longer than the red segment, equality if and only if a = b.

Click here for a GSP file to vary the lengths of a and b.

Construct your own GSP File and animation.

EXTENSION:

Consider adding an altitude to the right triangle as shown:

1. Find h in terms of a and b.

2. Show h is less than or equal to the geometric mean (the red segment) with equality if and only if a = b.

Click here for a GSP file.

Construct your own GSP file.

 

EXTENSION:

Given a rectangle with length a and width b, construct a square of the same area.