Apollonius Theorem

In mathematics, theorems are claims that have been proven based on previously established statements, such as theorems, or statements that are widely accepted, such as axioms. Theorems are mathematical results obtained from a collection of axioms that have been proven to be correct. Some mathematical axioms are called mathematical logic, but only when systems are expressed as questions or statements.

Theorems are mathematical assertions with proven results based on previously established claims like theorems and commonly accepted statements like axioms. The conclusions drawn from a set of axioms that have been proven to be accurate are known as theorems. In mathematics, where the axioms are of numerical logic and the systems are in the form of a question, this word is especially prevalent.

The Theorem of Apollonius

This is an elementary theorem that connects the length of a triangle’s median to the lengths of its edges. It is named after the Greek mathematician Apollonius. This claims that the sum of the squares of any two triangle’s sides is equal to twice the square on half of the third side, plus twice the square on the median that bisects the third side. In general, coordinate geometry is used to establish Apollonius’ theorem is valid, however, the Pythagorean theorem and vectors can also be used. Let’s have a look at the theorem’s formulation and proof. The length of a triangle’s median is proportional to the lengths of its sides, according to Apollonius’ theorem. While most of the world knows it as it is, the theorem is more often known in East Asia as Pappus’ theorem or the midpoint theorem. The Pythagorean theorem from the cosine rule, as well as vectors, can be used to establish it. The theorem is named after Apollonius of Perga, a Greek mathematician.

The Theorem of Apollonius 

The most essential collection of components in the geometry of triangles is without a doubt the median, which can be considered independent of the geometric shapes of the triangles. The sides and medians of a triangle are connected, according to Apollonius’ Theorem. The link between the lengths of a triangle’s sides and the length of its median is known as Apollonius’ theorem.

The Theorem of Apollonius

“The sum of squares on two sides of a triangle is equal to the sum of the squares on one half of the third side plus the sum of squares on the third side’s median.”

OR

Because the midpoint of any LMN triangle is O, the formula (LM)2+ (LN)2= 2 [(LO)2+(MO)2] = O is used.

Proof: Consider that O is the rectangular form’s origin, and MN and OY are the x- and y-axes of Cartesian coordinates, respectively. The coordinates of points M and N are (a, 0) and (-a, 0), respectively, when MN = 2a. If (b, c) are the L coordinates, then

LO2 = (c – 0)2 + (b – 0)2 since the coordinates of the point O are 0,0.

= b2 + c2;

In other sense

 MO2 = (0 – 0)² + (- a – 0)² = a²

LN² = (c – 0) ² + (b – a) ² = c² + (a – b)²

= 2c² + 2 (a² + b²)

= 2(b² + c²) + 2a²

= 2LO² + 2MO²

= 2 (LO² + MO²).

= 2(MO² + LO²).

it has been established. 

By the Pythagorean Theorem 

Statement: For the triangle let say ABC with M be the midpoint of its side BC, 

AB² + AC² = 2{AM² + (BC/2​) ²}

Proof: Let AH be the perpendicular from A on BC

So, it is clear that

BM = CM = BC/2

BH + CH = BC

Now, using Pythagorean Theorem

AB² = AH² + BH²

AC² = AH² + CH²

AM² = AH² + MH².​

From the above equations, we have:

AB² + AC² = 2AH² + BH² + CH²

      = 2AH² + 2MH² + BH² − MH² + CH² − MH²

      = 2AM² + (BH + MH) (BH−MH) + (CH+MH) (CH−MH)

      = 2AM² + (BH+MH) ⋅ BM + CM ⋅ (CH−MH)

      = 2AM² + BC²/2

      = 2 (AM² + (BC/2) ²). 

Thus proven.

Point To Remember

• Apollonius’ theorem is a theorem in geometry that relates the length of a triangle’s median to the lengths of its sides. “The sum of squares of any two sides of the given triangle equal to twice of  the square on half the third side, plus twice the square on the median bisecting the third side,” it explains.

Conclusion

Apollonius’ theorem is a theorem in geometry that relates the length of a triangle’s median to the lengths of its sides. “The sum of the squares of any two sides of any triangle equals twice the square on half of the third side, plus twice the square on the median bisecting the third side,” it says.