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>
- When in addition AB = AC, that is, the triangle is isosceles, the theorem reduces to the Pythagorean theorem,
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- <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 = 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\qquad (i)</math>
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.\qquad (ii)</math>
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>
- <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.
