Introduction
It is understood that theorems in mathematics refer to the statement of a result that has been proven based on previously set statements, such as theorems, or statements commonly acknowledged, such as axioms. In mathematics, theorems are defined as the results derived from a set of axioms that are found to be accurate. There are some axioms in mathematics that are considered mathematical logic, but with systems formulated as questions or statements.
Apollonius' Theorem
Named after the Greek mathematician Apollonius, this theorem is an elementary theorem that relates the length of a median of any triangle to the lengths of its edges. It states that the sum of squares of any two sides of a triangle is equal to twice of its square on half of the third side, along with the twice of its square on the median that bisects the third side. Apollonius' theorem, in general, is proved to be correct by using coordinate geometry, but it can also be proved by using the Pythagorean theorem and vectors. Now, let's go through the statement and proof of this theorem.
Statement and Proof of Apollonius' Theorem
There is no doubt that medians make up the most important set of components in the geometry of triangles and they can be regarded as independent of the geometric shapes of the triangles. According to Apollonius' Theorem, the sides and the medians of the triangle are related. Apollonius' theorem refers to the relationship between the lengths of the sides of a triangle and the length of its median.
Apollonius’ Theorem Statement
"The sum of the squares on two sides of a triangle equals the sum of the squares on one half of the third side, plus the sum of the squares along the median of the third side"
OR
The midpoint of any triangle LMN is O, so the formula (LM)²+ (LN)²= 2 \[\left [ (LO)² + (MO)² \right ]\] = O.
Apollonius’ Theorem Proof
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Consider that O is the origin of the rectangular form and MN and OY represent the x-axis and y-axis, respectively, of the Cartesian coordinates. When MN = 2a, the coordinates of points M, as well as N, are (a, 0) and (-a, 0), respectively. When L coordinates are (b, c), then
Since the coordinates of the point O are [0, 0], LO2 = (c – 0)2 + (b – 0)2 .
= c² + b²;
In other words, LM2 = (c – 0)² + (b + a)² = c² + (a + b)²
MO² = (0 – 0)² + (- a – 0)² = a²
LN² = (c – 0) ² + (b – a) ² = c² + (a – b)²
In other words, LN2 + LM2 = c2 + (a + b)² + c² + (b – a)²
= 2c² + 2 (a² + b²)
= 2(b² + c²) + 2a²
= 2LO² + 2MO²
= 2 (LO² + MO²).
= 2(MO² + LO²). Hence Proved.
Statement and Proof by the Pythagorean Theorem
Statement: For a triangle ABC with M be the midpoint of its side BC,
AB² + AC² = 2{AM² + (BC/2)²}
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Proof: Let AH be the perpendicular from A on BC
So, it is clear that
BM = CM = BC/2
BH + CH = BC
Now, use the Pythagorean Theorem
AB² = AH² + BH²
AC² = AH² + CH²
AM² = AH² + MH².
From the above equations, we can conclude that:
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). Hence, proved
Statement and Proof by Vectors
Statement: For a triangle, ABC having M as the midpoint of side BC, AB² + AC² = 2 (AM² + (BC/2)²), i.e., triangle ABC satisfies Apollonius's theorem by using vectors.
Proof: Let A be the Cartesian coordinate of triangle ABC and define AB = ∣b∣ and AC = ∣c∣, then it is clear that AM = (b+c)/2 and BC = ∣c∣ - ∣b∣
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AB² + AC² = ∣b∣²+ ∣c∣²
=1/2 (2∣b∣²+ 2∣c∣²)
= ½ (∣b∣² + ∣c∣² + 2b⋅c + ∣b∣² + ∣c∣²− 2b⋅c)
= ½ {(b+c)² + (c−b)²}
= ½ (4AM²+ BC2)
= 2 (AM² + (BC/2)²) Hence Proved.
FAQs on Apollonius Theorem
1. How is the Apollonius theorem used?
In elementary geometry, Apollonius' Theorem is comparable to Pythagoras' Theorem. A median of a triangle can be calculated to determine its length. This is equivalent to the Parallelogram Law.
2. Who discovered the Apollonius theorem?
The intersection of a plane with a cone at different angles produces conic sections or two-dimensional figures. Mathematicians of the ancient Greek world developed a thorough understanding of these figures, which has survived especially in works like those of Apollonius of Perga.
3. What was the reason for Apollonius renaming the conic sections?
In the Greek language, those words ellipsis, parabolic, and hyperbolic were known as parabolas, hyperbolas, and ellipses by Apollonius. It had a better way of defining the conic sections. Thus, he began to use a double-napped cone instead of a single-napped cone.
4. What did Apollonius write?
In Greek history, Apollonius is known as 'The Great Geometer' because he was a great mathematician. Mathematical terms such as parabola, ellipse, and hyperbola were introduced by his famous book, Conics.
5. What is the Apollonius Theorem Formula?
The Apollonius theorem gives the relationship between the median of a triangle and its sides in Mathematics. Alternatively, the theorem is known as the median of a triangle. According to this theorem, when any two sides of a triangle are squared separately, their sum will be equal to that which comes when two times the square of half the third side and two times the square of the median from the third side to the opposite vertex are added. When ABC is a triangle with sides AB, BC, and AC ordered in order, and AD is the median to BC, the following formula is used for the Apollonius theorem:
AB² + AC² = 2 (AD² + BD²)