geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, Apollonius's theorem is a

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

relating the length of a

median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...

of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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. The theorem is found as proposition VII.122 of

Pappus of Alexandria Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...

's ''Collection'' (). It may have been in

Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...

's lost treatise ''Plane Loci'' (c. 200 BC), and was included in Robert Simson's 1749 reconstruction of that work.

Statement and relation to other theorem

In any triangle

ABC,

AD

is a median (

, BD,  = , CD,

), then

, AB, ^2+, AC, ^2=2(, BD, ^2+, AD, ^2).

It is a special case of Stewart's theorem. For an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

with

, AB,  = , AC, ,

the median

AD

is perpendicular to

BC

and the theorem reduces to the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

for triangle

ADB

(or triangle

ADC

). From the fact that the diagonals of a

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

bisect each other, the theorem is equivalent to the

parallelogram law In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) 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 s ...

Proof

The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see

). The following is an independent proof using the law of cosines. Let the triangle have sides

a, b, c

with a median

d

drawn to side

a.

Let

m

be the length of the segments of

a

formed by the median, so

m

is half of

a.

Let the angles formed between

a

and

d

\theta

and

\theta^,

where

\theta

includes

b

and

\theta^

includes

c.

Then

\theta^

is the supplement of

\theta

and

\cos \theta^ = - \cos \theta.

The

law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...

for

\theta

and

\theta^

states that

\begin
b^2 &= m^2 + d^2 - 2dm\cos\theta \\
c^2 &= m^2 + d^2 - 2dm\cos\theta' \\
&= m^2 + d^2 + 2dm\cos\theta.\, \end

Add the first and third equations to obtain

b^2 + c^2 = 2(m^2 + d^2)

as required.

Statement and relation to other theorem

Proof

See also

References

Further reading