This article is about the lengths of the sides of a triangle. For his work on circles, see Problem of Apollonius.
In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. 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".
green/blue areas = red area
Pythagoras as a special case: green area = red area
Specifically, in any triangle if is a median, then
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.[1]
Let the triangle have sides with a median drawn to side Let be the length of the segments of formed by the median, so is half of Let the angles formed between and be and where includes and includes Then is the supplement of and The law of cosines for and states that
David B. Surowski: Advanced High-School Mathematics. p. 27
January 07, 2023
apollonius, theorem, this, article, about, lengths, sides, triangle, work, circles, problem, apollonius, geometry, theorem, relating, length, median, triangle, lengths, sides, states, that, squares, sides, triangle, equals, twice, square, half, third, side, to. This article is about the lengths of the sides of a triangle For his work on circles see Problem of Apollonius In geometry Apollonius s theorem is a theorem relating the length of a median of a triangle to the lengths of its sides 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 green blue areas red area Pythagoras as a special case green area red area Specifically in any triangle A B C displaystyle ABC if A D displaystyle AD is a median then A B 2 A C 2 2 A D 2 B D 2 displaystyle AB 2 AC 2 2 left AD 2 BD 2 right It is a special case of Stewart s theorem For an isosceles triangle with A B A C displaystyle AB AC the median A D displaystyle AD is perpendicular to B C displaystyle BC and the theorem reduces to the Pythagorean theorem for triangle A D B displaystyle ADB or triangle A D C displaystyle ADC From the fact that the diagonals of a parallelogram bisect each other the theorem is equivalent to the parallelogram law The theorem is named for the ancient Greek mathematician Apollonius of Perga Contents 1 Proof 2 See also 3 References 4 External linksProof Edit Proof of Apollonius s theorem 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 1 Let the triangle have sides a b c displaystyle a b c with a median d displaystyle d drawn to side a displaystyle a Let m displaystyle m be the length of the segments of a displaystyle a formed by the median so m displaystyle m is half of a displaystyle a Let the angles formed between a displaystyle a and d displaystyle d be 8 displaystyle theta and 8 displaystyle theta prime where 8 displaystyle theta includes b displaystyle b and 8 displaystyle theta prime includes c displaystyle c Then 8 displaystyle theta prime is the supplement of 8 displaystyle theta and cos 8 cos 8 displaystyle cos theta prime cos theta The law of cosines for 8 displaystyle theta and 8 displaystyle theta prime states thatb 2 m 2 d 2 2 d m cos 8 c 2 m 2 d 2 2 d m cos 8 m 2 d 2 2 d m cos 8 displaystyle begin aligned b 2 amp m 2 d 2 2dm cos theta c 2 amp m 2 d 2 2dm cos theta amp m 2 d 2 2dm cos theta end aligned Add the first and third equations to obtainb 2 c 2 2 m 2 d 2 displaystyle b 2 c 2 2 m 2 d 2 as required See also EditFormulas involving the medians lengths Line segment joining a triangle s vertex to the midpoint of the opposite sideReferences Edit Godfrey Charles Siddons Arthur Warry 1908 Modern Geometry University Press p 20 External links EditThree Proofs of Apollonius Theorem by HC Rajpoot from Academia eduApollonius Theorem at PlanetMath David B Surowski Advanced High School Mathematics p 27 Retrieved from https en wikipedia org w index php title Apollonius 27s theorem amp oldid 1126699116, wikipedia, wiki, book, books, library,