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Apollonius' theorem
From Wikipedia, the free encyclopedia
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 mBD = nDC), then
- mAB2 + nAC2 = mBD2 + nDC2 + (m + n)AD2.
- When m = n( = 1), that is, AD is the median falling on BC, the theorem reduces to
- When in addition AB = AC, that is, the triangle is isosceles, the theorem reduces to the Pythagorean theorem,
In simpler words, in any triangle 
, if 
is a median, then 
= 
To prove this theorem, let 
be a perpendicular dropped on 
from the point 
. Then, in the right-angled triangles 
and 
, by Pythagoras' Theorem, we have 
=
=
...........(i)
and
=
=
...........(ii)
Adding equations (i) and (ii),
=
=
{since 
,thus 
}
=
= {since 
is a right angle}
And thus the theorem is proved.