Apollonius' theorem

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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

mAB2 + nAC2 = mBD2 + nDC2 + (m + n)AD2.

[edit] 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.DX\qquad (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.


[edit] See also