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In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".

green/blue areas = red area

Specifically, in any triangle ABC, if AD is a median, then

$${\displaystyle |AB|^{2}+|AC|^{2}=2(|AD|^{2}+|BD|^{2}).}$$

It is a special case of Stewart's theorem. For an isosceles triangle with |AB| = |AC|, the median AD is perpendicular to BC and the theorem reduces to the Pythagorean theorem for triangle ADB (or triangle ADC). From the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.

The theorem is named for the ancient Greek mathematician Apollonius of Perga.

Pythagoras as a special case:
green area = red area

Proof

Proof of Apollonius's theorem

The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines.[1]

Let the triangle have sides a, b, c with a median d drawn to side a. Let m be the length of the segments of a formed by the median, so m is half of a. Let the angles formed between a and d be θ and θ′, where θ includes b and θ′ includes c. Then θ′ is the supplement of θ and cos θ′ = −cos θ. The law of cosines for θ and θ′ states that

\begin{align} b^2 &= m^2 + d^2 - 2dm\cos\theta \\ c^2 &= m^2 + d^2 - 2dm\cos\theta' \\ &= m^2 + d^2 + 2dm\cos\theta.\, \end{align}

Add the first and third equations to obtain

$${\displaystyle b^{2}+c^{2}=2(m^{2}+d^{2})}$$

as required.
References

Godfrey, Charles; Siddons, Arthur Warry (1908). Modern Geometry. University Press. p. 20.

Apollonius Theorem at PlanetMath.org.
David B. Surowski: Advanced High-School Mathematics. p. 27

vte

Ancient Greek and Hellenistic mathematics (Euclidean geometry)
Mathematicians
(timeline)

Treatises

Almagest Archimedes Palimpsest Arithmetica Conics (Apollonius) Catoptrics Data (Euclid) Elements (Euclid) Measurement of a Circle On Conoids and Spheroids On the Sizes and Distances (Aristarchus) On Sizes and Distances (Hipparchus) On the Moving Sphere (Autolycus) Euclid's Optics On Spirals On the Sphere and Cylinder Ostomachion Planisphaerium Sphaerics The Quadrature of the Parabola The Sand Reckoner

Problems

Angle trisection Doubling the cube Squaring the circle Problem of Apollonius

Concepts/Definitions

Circles of Apollonius
Apollonian circles Apollonian gasket Circumscribed circle Commensurability Diophantine equation Doctrine of proportionality Golden ratio Greek numerals Incircle and excircles of a triangle Method of exhaustion Parallel postulate Platonic solid Lune of Hippocrates Quadratrix of Hippias Regular polygon Straightedge and compass construction Triangle center

Results
In Elements

Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon

Apollonius

Apollonius's theorem

Other

Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Menelaus's theorem Pappus's area theorem Ptolemy's inequality Ptolemy's table of chords Ptolemy's theorem Spiral of Theodorus

Centers

Cyrene Library of Alexandria Platonic Academy

Other

Ancient Greek astronomy Greek numerals Latin translations of the 12th century Neusis construction