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## Apollonius Theorem

The theorem states that for every two circles there is a unique line that passes through their centers and is perpendicular to their common diameter. In mathematics, the Apollonius theorem is a statement in plane geometry that states that a circle is the locus of points equidistant from two other points. The theorem is attributed to the Greek mathematician Apollonius of Perga.

## Statement and Proof of Apollonius’ Theorem

Apollonius’ theorem states that for every three points in a plane there exists a unique circle that passes through all three points.

**Proof: **We will use proof by contradiction. Suppose that there exists a circle that passes through all three points but that is not unique. This would mean that there are two circles that pass through all three points. But then the points would not be unique, which is a contradiction. Therefore, there must be a unique circle that passes through all three points.

## Apollonius’ Theorem Statement

Let ABC be a right triangle with right angle C. If a and b are the lengths of the other two sides, then Apollonius’ theorem states that

**a2 + b2 = c2.**

## Apollonius’ Theorem Proof

Let \(A\) and \(B\) be points on a circle and let \(P\) be the point on the circle that is the midpoint of \(AB\).

Then, \(AP = PB\).

## Statement and Proof by the Pythagorean Theorem

A right triangle has the length of its longest side, or hypotenuse, as its hypotenuse. The other two sides are the short side and the long side. The Pythagorean theorem states that the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse. This theorem is represented by the equation: a^2 + b^2 = c^2.

To prove the Pythagorean theorem, we can use a right triangle that has been constructed out of unit squares. The length of the short side is 1 unit, the length of the long side is 2 units, and the length of the hypotenuse is 3 units. We can see that the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse. 1^2 + 2^2 = 3^2.

## Statement and Proof by Vectors

Statement: If vectors a and b are both nonzero, then the dot product a · b is positive.

Proof:

We will show that the dot product a · b is positive if and only if a and b are both nonzero.

If a and b are both nonzero, then the dot product a · b is positive.

## Examples

Apollonius’ theorem is a fundamental theorem in geometry that states the following: if the perpendicular bisectors of the sides of a triangle intersect at a point, then the angles of the triangle are equal. This theorem is named after the ancient Greek mathematician Apollonius of Perga, who first proved it in the third century B.C.

Apollonius’ theorem can be used to prove many other properties of triangles. For example, it can be used to prove that the circumcenter of a triangle is the point at which the perpendicular bisectors of its sides intersect. This point is also the center of the circumcircle, which is the circle that passes through all of the triangle’s vertices.

The theorem can also be used to prove that the incenter of a triangle is the point at which the angle bisectors of its angles intersect. The incenter is the center of the incircle, which is the circle inscribed inside the triangle and tangent to each of its sides.

Apollonius’ theorem can also be used to prove that the orthocenter of a triangle is the point at which its altitudes intersect. An altitude of a triangle is a line that passes through a vertex and is perpendicular to the opposite side. The orthocenter is the point at which the three altitudes of a triangle intersect.

## FAQs

Q: What is Apollonius Theorem?

A: Apollonius Theorem is a geometric theorem that relates the length of a median of a triangle to the lengths of its other sides.

Q: How can Apollonius Theorem be stated?

A: Apollonius Theorem states that in a triangle, the sum of the squares of the lengths of the two smaller sides is equal to twice the square of the length of the median that bisects the third side.

Q: How can Apollonius Theorem be proved?

A: One way to prove Apollonius Theorem is to use the Law of Cosines. Suppose a triangle ABC has sides of length a, b, and c, and that the median to side c has length m. Then, we can use the Law of Cosines to express m in terms of a, b, and c. Then, we can square both sides of this equation and use the Law of Cosines again to simplify the expression. The resulting equation is Apollonius Theorem.

Q: What are some applications of Apollonius Theorem?

A: Apollonius Theorem is useful in geometry and trigonometry. It can be used to solve problems involving triangles, such as finding the length of a median or determining whether a triangle is acute, right, or obtuse.

Q: How can Apollonius Theorem be used to find the length of a median?

A: To use Apollonius Theorem to find the length of a median, we first identify the triangle’s sides and the median in question. Then, we plug the lengths of the two smaller sides and the median into Apollonius Theorem and solve for the length of the median.