In a right triangle, the square of the altitude is equal to the product of the two segments that it divides the base into.
In other words, the altitude is equal to the square root of the product of these two segments, which means that it is their geometric mean. Now, let's draw two line segments of the length x and y.
To construct their geometric mean we need to draw them so that they form a single segment AB. It will be the hypotenuse of a right triangle that we will later construct.
Now, we need to construct a midpoint of the segment by drawing a perpendicular bisector of AB. First, we will put the compass point on A so that the opening of the compass must be greater than 12AB and draw an arc. Then, with the same compass setting we will put the compass point on point B and draw another arc.
Next we should draw a line through their points of intersection. As we mentioned before, it is a perpendicular bisector and it intersects AB at the midpoint, which we can label as M.
Let's now draw a semicircle with the center at M. To do this we will put the compass point on M, open the compass to AM, and draw a semicircle.
Next, we need to draw a perpendicular line to segment AB through point C. We will name its point of intersection with the semicircle D.
This point will be the third vertex of the triangle. Let's draw segments AD and BD and form this triangle.
Now, let's recall the Corollary 2 of the Inscribed Angle Theorem, which states the following.
Corollary 2
An angle inscribed in a semicircle is a right angle.
Angle ∠ ADB is inscribed in a semicircle, so it is a right angle. Therefore, △ ADB is a right triangle and DC is its altitude. By the Right Triangle Altitude Theorem, we conclude that DC is a geometric mean of x and y.
