A wonderful Proof of $A.M \geq G.M$ inequality

Lets us consider a semicircle with diameter AB. If C is any point on the semicircle, then ABC forms a right angled triangle with right angle at C .

Let CD be the perpendicular to AB dropped from C . Let AD= a and DB=b

The trianges DAC and DCB are similar , hence

$\frac{DA}{DC} = \frac{DC}{DB}$

therefore

$DC^{2} = DA . DB = ab$

and thus $DC = \sqrt{ab}$

if r is the radius of the semicircle , then $DC \leq r$

we know $r = \frac{diameter}{2} = \frac{a + b}{2}$

Thus

$\boxed{ \frac{a + b}{2} \geq \sqrt{ab}}$

If $DC = r$ that is if the Point D comes on the center of the circle then $a = b$

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