Apollonius's theorem: Difference between revisions
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Corrected an error. Reduces to Pythag theorem in case of right-angled triangle, not isoceles. |
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The theorem is named for [[Apollonius of Perga]]. |
The theorem is named for [[Apollonius of Perga]]. |
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==Proof== |
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[[image:ApolloniusTheoremProof.svg|left|thumb|Proof of Apollonius' theorem]] |
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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.<ref>Following Godfrey & Siddons</ref> |
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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 |
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:<math> |
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\begin{align} |
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b^2 &= m^2 + d^2 - 2dm\cos\theta \\ |
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c^2 &= m^2 + d^2 - 2dm\cos\theta' \\ |
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&= m^2 + d^2 + 2dm\cos\theta.\, \end{align} |
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</math> |
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Add these equations to obtain |
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:<math>b^2 + c^2 = 2m^2 + 2d^2\,</math> |
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as required. |
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==References== |
==References== |
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Revision as of 13:32, 15 September 2014
In geometry, Apollonius' 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
It is a special case of Stewart's theorem. For a right-angled triangle the theorem reduces to the Pythagorean theorem. From the fact that diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.
The theorem is named for Apollonius of Perga.
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References
- Godfrey, Charles; Siddons, Arthur Warry (1908). Modern Geometry. University Press. p. 20.
- Apollonius Theorem at PlanetMath.