What Is Apollonius Theorem?

In triangle, ABC,

The Apollonius theorem relates the length of a median of a triangle to the lengths of its side.

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Apollonius theorem

Let \$a,b,c\$ the sides of a triangle and \$m\$ the length of the median to the side with length \$a\$ . Then \$b^2+c^2=2m^2+frac{a^2}{2}\$ .

If \$b=c\$ (the triangle is isosceles), then the theorem reduces to the Pythagorean theorem, \$\$ m^2 + (a/2)^2 = b^2. \$\$

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In geometry, Apollonius' theorem is a theorem relating the length of a median of a triangle to the lengths of its side. Specifically, in any triangle ABC, if AD is a median, then

$AB^2 + AC^2 = 2(AD^2+BD^2),$

It is a special case of Stewart's theorem. For an isosceles triangle the theorem reduces to the Pythagorean theorem. From the fact that diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.

The theorem is named for Apollonius of Perga.

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 with a median d drawn to side a. Let m be the length of the segments of a formed by the median, so m is half of a. Let the angles formed between a and d be θ and θ′ where θ includes b and θ′ includes c. Then θ′ is the supplement of θ and cos θ′ = −cos θ. The law of cosines for θ and θ′ states

nbegin{align}nb^2 &= m^2 + d^2 - 2dmcostheta nc^2 &= m^2 + d^2 - 2dmcostheta' n&= m^2 + d^2 + 2dmcostheta., end{align}n
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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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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.

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+p

2

)=b

2

m+c

2

n

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