Yahoo Answers: Answers and Comments for Can Some on give me a proof of apollonius theorem? [Homework Help]
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From Anonymous
enUS
Fri, 17 Aug 2012 00:40:17 +0000
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Yahoo Answers: Answers and Comments for Can Some on give me a proof of apollonius theorem? [Homework Help]
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From Jalol: I know two ways of proving. The first is based...
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Fri, 17 Aug 2012 01:35:24 +0000
I know two ways of proving. The first is based on cosine theorem. Choose any angle of triangle and express side and mediana using cosine theorem. Then equal cosines:
m^2=AB^2+(AC/2)^2  2AB*AC/2*cosA
BC^2=AB^2+AC^2  2*AB*AC*cosA
2)Using the theorem about parallelogram. (a^2+b^2)=d1^2+d2^2, where d1 and d2 are diagonals of par.
If you continue mediana until it's twice of its length and connect the end of mediana with two other vertexes you'll get a parallelogramm. One of its diagonals equals 2m.

From Learner: i) Apollonius Theorem:
"In a triangle A...
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Fri, 17 Aug 2012 22:38:33 +0000
i) Apollonius Theorem:
"In a triangle ABC, if AD is the median to BC, then AB² + AC² = 2(AD² + BD²)"
ii) As I am not sure of your knowledge on Cosine Law of triangle, let me provide the solution using basic geometrical concepts applied in high school geometry.
iii) Draw an altitude from vertex A to BC, to meet BC at E. [E may be within B & D or withing D & C, depends upon AB < AC or AB > AC; let us here take AB < AC, so E is between B & D]
iv) Applying Pythagoras theorem to respective right triangles,
From right triangle, AEB, AB² = AE² + BE²  (1)
From right triangle, AED, AD² = AE² + ED²  (2)
(1)  (2): AB²  AD² = BE²  ED² = (BD  DE)²  ED² = BD²  2BD*DE + ED²  ED²
==> AB²  AD² = BD²  2BD*DE  (3)
v) Similarly considering the right triangles, AEC & AED, we get,
AC²  AD² = DE²  ED² = (CD + DE)²  DE² = CD² + 2CD*DE + DE²  DE²
==> AC²  AD² = CD² + 2CD*DE = BD² + 2BD*DE  (4)
[Since AD is the median, D is mid point of BC; hence BD = DC]
vi) Adding (3) & (4)
==> AB² + AC²  2AD² = 2BD²
==> AB² + AC² = 2(AD² + BD²) [Proved]

From Will H: http://www.easycalculation.com/theorems/stewar...
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Fri, 17 Aug 2012 01:32:34 +0000
http://www.easycalculation.com/theorems/stewartorapollonius.php

From touchette: Apollonius Theorem
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Fri, 11 Nov 2016 17:22:34 +0000
Apollonius Theorem