In geometry, **Apollonius's theorem** is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".

Specifically, in any triangle if is a median, then

It is a special case of Stewart's theorem. For an isosceles triangle with the median is perpendicular to and the theorem reduces to the Pythagorean theorem for triangle (or triangle ). From the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.

The theorem is named for the ancient Greek mathematician 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.^{ [1] }

Let the triangle have sides with a median drawn to side Let be the length of the segments of formed by the median, so is half of Let the angles formed between and be and where includes and includes Then is the supplement of and The law of cosines for and states that

Add the first and third equations to obtain

as required.

A **circle** is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

A **quadrilateral** is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include **quadrangle**, **tetragon**, and **4-gon**. A quadrilateral with vertices , , and is sometimes denoted as .

In mathematics, the **trigonometric functions** are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

In geometry, a **solid angle** is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the *apex* of the solid angle, and the object is said to *subtend* its solid angle from that point.

In Euclidean geometry, a **parallelogram** is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

In Euclidean geometry, **Brahmagupta's formula** is used to find the area of any cyclic quadrilateral given the lengths of the sides.

In Euclidean geometry, a **cyclic quadrilateral** or **inscribed quadrilateral** is a quadrilateral whose vertices all lie on a single circle. This circle is called the *circumcircle* or circumscribed circle, and the vertices are said to be *concyclic*. The center of the circle and its radius are called the *circumcenter* and the *circumradius* respectively. Other names for these quadrilaterals are **concyclic quadrilateral** and **chordal quadrilateral**, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a **trapezium** in English outside North America, but as a **trapezoid** in American and Canadian English. The parallel sides are called the *bases* of the trapezoid and the other two sides are called the *legs* or the lateral sides. A *scalene trapezoid* is a trapezoid with no sides of equal measure, in contrast with the special cases below.

In geometry, **Thales's theorem** states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's *Elements*. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

In mathematics, the simplest form of the **parallelogram law** belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the sides: *AB*, *BC*, *CD*, *DA*. But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, *AB* = *CD* and *BC* = *DA*, the law can be stated as

The **haversine formula** determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the **law of haversines**, that relates the sides and angles of spherical triangles.

The **Pythagorean trigonometric identity**, also called simply the **Pythagorean identity**, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

In Euclidean geometry, **Ptolemy's theorem** is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

In geometry, the area enclosed by a circle of radius r is π*r*^{2}. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.1416.

In spherical trigonometry, the **law of cosines** is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.

In geometry, **Stewart's theorem** yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746.

In trigonometry, the **law of cosines** relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states

In mathematics, the **Pythagorean theorem**, or **Pythagoras's theorem**, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides *a*, *b* and *c*, often called the **Pythagorean equation**:

- ↑ Godfrey, Charles; Siddons, Arthur Warry (1908).
*Modern Geometry*. University Press. p. 20.

- Apollonius Theorem at PlanetMath .
- David B. Surowski:
*Advanced High-School Mathematics*. p. 27

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