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Apollonius’ Theorem

 

Theorems are assertions in mathematics with proven results based on previously set claims, such as theorems, and generally confirmed statements, such as axioms. Theorems are defined as the outcomes that are shown to be correct based on a collection of distinct axioms. This phrase is most commonly used in mathematics, where the axioms are of numerical logic with systems in the form of questions. A line segment traced from a vertex to the middle of the opposite side of the vertex is referred to as a triangle's median. At a given location, the medians of a triangle are contemporaneous. The point of convergence is referred to as the centroid.

Statement and Proof of Apollonius’ Theorem

Apollonius of Perga, a Greek mathematician, is commemorated by the theorem's name. Medians are recognized to create the most significant sets of elements in the geometries of triangles that are strongly connected to the triangle despite the geometric forms. The relationship seen between medians and the triangle sides is recognized in Apollonius' Theorem. Apollonius' theorem is a type of hypothesis that relates the size of a triangle's median to the length of its sides. Apollonius' theorem is a fundamental geometry theorem that associates the extent of a triangle's median to the dimensions of its sides. While most of the world alludes to it as it is, in the Eastern part of Asia, the postulate is well known as Pappus' theorem or the midpoint theorem. It may be demonstrated using the Pythagorean theorem and vectors, as well as the cosine rule.

Apollonius’ Theorem Statement:

“The aggregate of the squares of any two dimensions of a triangle equals twice its square on half of the third side, adding double its square on the median which bisects the third length,” says the statement.

Or

If Q is at the centre of the line ST, one of the lengths of the triangle (RST), then prove that RS² + RT² = 2 {SQ² + RQ²}.

Apollonius’ Theorem Proof:

Select the beginning of the rectangular type of the Cartesian coordinates at the point Q and the x-axis moving along the lengths of ST and also QY as y – axis. If in case ST = 2i, then the coordinates of the points T, as well as S, are defined as (i, 0) and (- i, 0) respectively. If coordinates of the point ‘R’ are (j, k), then

RQ² = (k – 0)² + (j – 0) ², ({0, 0} are the coordinates of the point Q)

= k² + j²;

RS² = (k – 0)² + (j + i) ² = k² + (i + j)²

SQ² = (0 – 0)² + (- i – 0) ² = i²

also, RT² = (k – 0) ² + (j – i) ² = k² + (i – j)²

Therefore, RT² + RS² = k² + (i + j)² + k² + (j – i)²

= 2k² + 2 (i² + j²)

= 2(j² + k²) + 2i²

= 2RQ² + 2SQ²

= 2 (RQ² + SQ²).

= 2(SQ² + RQ²). {Therefore, Proved}

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