Last February, we have used area to prove the Arithmetic Mean – Geometric Mean Inequality (AM-GM Inequality). In this post, we show a simpler proof.
Recall that the AM-GM Inequality states that given two numbers, their geometric mean is always less than or equal to their arithmetic mean. That is given two numbers and ,
Consider the semicircle above with radius . If we construct any vertical segment from to a point on the semicircle, we are sure that its length is less than or equal to the length of radius. That is, we are sure that the following inequality always holds
The diameter of the circle is equal to , so its radius
If we draw and , we form two right triangles that are similar, the triangle containing the sides , , and , and the triangle containing the sides , , and . In effect, since the two triangles are similar,
which means that and
Now, since ,
and that proves the claim above.