360
MATHEMATICS MAGAZINE
The Parallelogram with Maximum Perimeter
for Given Diagonals Is the Rhombus—A Proof
Without Words and a Corollary
ANGEL PLAZA
Universidad de Las Palmas de Gran Canaria
Las Palmas, Spain
[email protected]
Theorem. The parallelogram with maximum perimeter for given diagonals is the
rhombus.
Proof.
P ( π / 2)
P ( α)
b
b
B
α
a
A
a
a 2 + b2 − 2ab cos α + a 2 + b2 + 2ab cos α
≤ 2 a 2 + b2
|AP(α)| + |BP(α)| =
(by the Law of Cosines and the arithmetic mean-root mean square inequality).
Corollary. For two positive numbers a and b, let us denote by A, G, and R,
respectively their arithmetic mean, geometric mean, and root mean square, then
2A ≥ R + G.
c Mathematical Association of America
Math. Mag. 88 (2015) 360–361. doi:10.4169/math.mag.88.5.360. MSC: Primary 97G40.
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All use subject to JSTOR Terms and Conditions
361
VOL. 88, NO. 5, DECEMBER 2015
Proof.
R
A
A
a
G
b
(where two parallelograms with equal diagonals have been constructed from the geometric representation of the arithmetic mean, geometric mean, and root mean square
of two positive given numbers).
Summary. By the Law of Cosines and the arithmetic mean-root mean square inequality it is proved without
words that The Parallelogram with Maximum Perimeter for given Diagonals is the Rhombus. As a corollary it
also proved that for two positive numbers, their arithmetic mean is greater or equal than the arithmetic mean of
their geometric mean and their root mean square.
ANGEL PLAZA (MR Author ID: 350023) received his masters degree from Universidad Complutense de
Madrid in 1984 and his Ph.D. from Universidad de Las Palmas de Gran Canaria in 1993, where he is a Full
Professor in Applied Mathematics. He is interested in mesh generation and refinement, combinatorics and visualization support in teaching and learning mathematics.
Artist Spotlight
George Hart
Dragonflies, George Hart;
wood, 7.5 inches in diameter, 2009; private collection.
Constructed from 12 identical plywood components.
See interview on page
374.
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All use subject to JSTOR Terms and Conditions