Apollonius' theorem

In elementary geometry, Apollonius' theorem is a theorem relating several elements in a triangle.

It states that given a triangle "ABC", if "D" is any point on "BC" such that it divides "BC" in the ratio "n":"m" (or $mBD\; =\; nDC$), then

:$mAB^2\; +\; nAC^2\; =\; mBD^2\; +\; nDC^2\; +\; (m+n)AD^2.$

Special cases of the theorem

* When $m\; =\; n\; (=1)$, that is, "AD" is the median falling on "BC", the theorem reduces to

::$AB^2\; +\; AC^2\; =\; BD^2\; +\; DC^2\; +\; 2AD^2.\; ,$

* When in addition "AB" = "AC", that is, the triangle is isosceles, the theorem reduces to the Pythagorean theorem,

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

In simpler words, in any triangle $ABC,$, if $AD,$ is a median, then $AB^2\; +\; AC^2\; =\; 2(AD^2+BD^2),!$

To prove this theorem, let $AX,$' be a perpendicular dropped on $BC,$ from the point "'$A,$. Then, in the right-angled triangles $ABX,$ and $ACX,$, by Pythagoras' theorem, we have

: $AB^2\; =\; AX^2\; +\; BX^2,$

: $=\; AX^2\; +\; (BD+DX)^2,$

: $=\; AX^2\; +\; BD^2\; +\; DX^2\; +\; 2.BD.DXqquad\; (i)$

and

: $AC^2\; =\; AX^2\; +\; CX^2,$

: $=\; AX^2\; +\; (CD-DX)^2,$

: $=\; AX^2\; +\; CD^2\; +\; DX^2\; -\; 2.CD.DX.qquad\; (ii)$

Adding equations ("i") and ("ii"),

: $AB^2\; +\; AC^2,!$

: $=\; AX^2\; +\; BD^2\; +\; DX^2\; +\; 2.BD.DX\; +\; AX^2\; +\; CD^2\; +\; DX^2\; -\; 2.CD.DX,!$

: $=\; 2(AX^2\; +\; DX^2\; +\; BD^2),$

since $BD=DC,,$

: $2.BD.DX=2.DC.DX,!$

: $=\; 2(AX^2\; +\; DX^2)\; +\; 2BD^2,!$

: $=\; 2(AD^2\; +\; BD^2),!$

since $AXD,$ is a right angle"

And thus the theorem is proved.

ee also

* Stewart's theorem
* Parallelogram law
* Pythagorean theorem
* Menelaus' theorem
* Ceva's theorem