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
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case ...
of
Stewart's theorem. For an
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
with
the median
is perpendicular to
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 ...
for triangle
(or triangle
). From the fact that the diagonals of a
parallelogram bisect each other, the theorem is
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equival ...
to the
parallelogram law.
The theorem is named for the ancient Greek mathematician
Apollonius of Perga
Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
.
Proof
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.
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.
See also
*
References
External links
Three Proofs of Apollonius Theorem by HC Rajpootfrom
Academia.edu
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*
* David B. Surowski
''Advanced High-School Mathematics'' p. 27
{{Ancient Greek mathematics
Euclidean geometry
Articles containing proofs
Theorems about triangles