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. • 5

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.

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 • -1

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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