Apollonius' theorem

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

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

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.^[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

\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

↑ Following Godfrey & Siddons

Godfrey, Charles; Siddons, Arthur Warry (1908). Modern Geometry. University Press. p. 20.
Apollonius Theorem at PlanetMath.org.

This article is issued from Wikipedia - version of the 10/21/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.