See AM-GM Inside a Semicircle

Split a diameter into lengths a and b. The altitude from the split point to the semicircle has length √(ab), while the radius is (a+b)/2. An altitude inside the semicircle cannot exceed the radius.

Learning goalExplain geometrically why √(ab) ≤ (a+b)/2 and identify equality at a=b.

√ab ≤ a+b2

Step 1

Move the idea

Drag the slider

Try it: change one value

Move the slider

Change one parameter and watch what changes with it.

Step 2 · The aha

The inequality becomes a picture: geometric mean is an altitude; arithmetic mean is the radius.

The geometry

The right-triangle altitude theorem gives h²=ab, so h=√(ab). The semicircle radius is half the diameter, (a+b)/2.

Why equality is special

The altitude reaches the full radius only at the center of the diameter. That is exactly the balanced case a=b.

A proof you can remember

Instead of memorizing an algebraic manipulation, remember one picture: altitude ≤ radius.

Step 3 · One-question check

Did it click?

When does equality √(ab)=(a+b)/2 occur in the picture?

Continue the path

Back to the Visual Geometry learning path