Apollonius' theorem

This article is about the lengths of the sides of a triangle. For his work on circles, see Problem of Apollonius.

File:Apollonius' theorem.svg

Area of Green + Area of Blue = Area of Red

In geometry, Apollonius' theorem is a theorem relating the length of a median of a triangle to the lengths of its side. 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"
Specifically, in any triangle ABC, if AD is a median, then

$AB^2 + AC^2 = 2(AD^2+BD^2).\,$

It is a special case of Stewart's theorem. For a right-angled triangle the theorem reduces to the Pythagorean theorem. From the fact that diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.

The theorem is named for Apollonius of Perga.

Proof

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Proof of Apollonius' 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.

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

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

Add these equations to obtain

$b^2 + c^2 = 2(m^2 + d^2)\,$

as required.

References

• Apollonius Theorem at PlanetMath.org.