The Arithmetic-Geometric Mean Inequality (II)

​
a
1.5
b
3
The arithmetic-geometric mean inequality:
ab
≤
a+b
2
, for
a,b≥0
.
The yellow and green triangles are similar, so
b
L
=
L
a
. Therefore
2
L
=ab
,
L=
ab
, and the figure shows that
L≤
a+b
2
.

Details

Reference: C. D. Gallant, "Proof without Words: A Truly Geometric Inequality," Mathematics Magazine, 50(2), 1977 p. 98.

External Links

Arithmetic Mean (Wolfram MathWorld)
Geometric Mean (Wolfram MathWorld)

Permanent Citation

Soledad Mª Sáez Martínez, Félix Martínez de la Rosa
​
​"The Arithmetic-Geometric Mean Inequality (II)"​
​http://demonstrations.wolfram.com/TheArithmeticGeometricMeanInequalityII/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011