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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 <math>mBD = nDC</math>), then

<math>mAB^2 + nAC^2 = mBD^2 + nDC^2 + (m+n)AD^2.</math>

Special cases of the theorem

• When <math>m = n (=1)</math>, that is, AD is the median falling on BC, the theorem reduces to

<math>AB^2 + AC^2 = BD^2 + DC^2 + 2AD^2.\,\!</math><math>AB^2 + AC^2 = BD^2 + DC^2 + 2AD^2.\,\!</math>

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

<math> AD^2 + BD^2 = AB^2 (= AC^2).\,\!</math>

In simpler words, in any triangle <math>ABC\,\!</math>,
if <math>AD\,\!</math> is a median,
then
<math>AB^2 + AC^2\,\!</math>=
<math>2(AD^2+BD^2)\,\!</math>
To prove this theorem,
let <math>AX\,\!</math> be a perpendicular dropped on <math>BC\,\!</math>
from the point <math>A\,\!</math>.
Then, in the right-angled triangles <math>ABX\,\!</math> and <math>ACX\,\!</math>, by Pythagoras’ Theorem, we have
<math>AB^2 = AX^2 + BX^2\,\!</math>

=<math>AX^2 + (BD+DX)^2\,\!</math>

=<math>AX^2 + BD^2 + DX^2 + 2.BD.DX\,\!</math> ………..(i)

and

<math>AC^2 = AX^2 + CX^2\,\!</math>

=<math>AX^2 + (CD-DX)^2\,\!</math>

=<math>AX^2 + CD^2 + DX^2 - 2.CD.DX\,\!</math> ………..(ii)

Adding equations (i) and (ii),

<math>AB^2 + AC^2\,\!</math>

=<math>AX^2 + BD^2 + DX^2 + 2.BD.DX + AX^2 + CD^2 + DX^2 - 2.CD.DX\,\!</math>

=<math>2(AX^2 + DX^2 + BD^2)\,\!</math>
{since <math>BD=DC\,\!</math>,thus <math>2.BD.DX=2.DC.DX\,\!</math>}

=<math>2(AX^2 + DX^2) + 2BD^2\,\!</math>

=<math>2(AD^2 + BD^2)\,\!</math> {since <math>AXD\,\!</math> is a right angle}

And thus the theorem is proved.

See also

• Stewart’s theorem
• Parallelogram law
• Pythagorean theorem
• Menelaus’ theorem
• Ceva’s theorem

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