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.

Stewart's or Apollonius Theorem



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


From this we conclude that,
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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

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