Apollonius's theorem







green/blue areas = red area




Pythagoras as a special case:
green area = red area


In geometry, Apollonius's 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).{displaystyle |AB|^{2}+|AC|^{2}=2(|AD|^{2}+|BD|^{2}).}{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.



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


b2=m2+d2−2dmcos⁡θc2=m2+d2−2dmcos⁡θ′=m2+d2+2dmcos⁡θ.{displaystyle {begin{aligned}b^{2}&=m^{2}+d^{2}-2dmcos theta \c^{2}&=m^{2}+d^{2}-2dmcos theta '\&=m^{2}+d^{2}+2dmcos theta .,end{aligned}}}<br />
begin{align}<br />
b^2 &= m^2 + d^2 - 2dmcostheta \<br />
c^2 &= m^2 + d^2 - 2dmcostheta' \<br />
&= m^2 + d^2 + 2dmcostheta., end{align}<br />

Add these equations to obtain


b2+c2=2(m2+d2){displaystyle b^{2}+c^{2}=2(m^{2}+d^{2})}{displaystyle b^{2}+c^{2}=2(m^{2}+d^{2})}

as required.



References





  1. ^ Godfrey, Charles; Siddons, Arthur Warry (1908). Modern Geometry. University Press. p. 20..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}




External links




  • Apollonius Theorem at PlanetMath.org.

  • David B. Surowski: Advanced High-School Mathematics. p. 27




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