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Stewart's or Apollonius Theorem Proof

The theorem states the the relation between the length of sides of a triangle and the segment's length from a vertex to a point on the opposite side.This is also referred as Apollonius Theorem.

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Stewart's or Apollonius Theorem

Diagram:

Proof:

Let be the angle

Applying cosine's law on triangle AXB, we get
and so,

Applying the cosine's law on triangle AXC,
we get

and thus we get ,


From the above expressions we obtain,


By cancelling 2p on both sides and collecting, the equation can be obtained as,

From above equation we consider that

Where,a=m+n

From this we conclude that,
a(mn+p2)=b2m+c2n

  
   
  
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The triangle's cevian length and the length of the sides can be obtained from the proof of Stewart's Theorem