![Automedian triangle](https://upload.wikimedia.org/wikipedia/commons/9/92/Automedian_triangle.svg)
In
plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, an automedian triangle is a
triangle
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 \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
in which the lengths of the three
medians (the line segments connecting each
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
to the midpoint of the opposite side) are proportional to the lengths of the three sides, in a different order. The three medians of an automedian triangle may be
translated
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
to form the sides of a second triangle that is
similar to the first one.
Characterization
The side lengths of an automedian triangle satisfy the formula 2''a''
2 = ''b''
2 + ''c''
2 or a permutation thereof, analogous to the
Pythagorean theorem characterizing
right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
s as the triangles satisfying the formula ''a''
2 = ''b''
2 + ''c''
2.
That is, in order for the three numbers ''a'', ''b'', and ''c'' to be the sides of an automedian triangle, the sequence of three squared side lengths ''b''
2, ''a''
2, and ''c''
2 should form an
arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
.
[.]
Construction from right triangles
If ''x'', ''y'', and ''z'' are the three sides of a right triangle, sorted in increasing order by size, and if 2''x'' < ''z'', then ''z'', ''x'' + ''y'', and ''y'' − ''x'' are the three sides of an automedian triangle. For instance, the right triangle with side lengths 5, 12, and 13 can be used to form in this way an automedian triangle with side lengths 13, 17, and 7.
[.]
The condition that 2''x'' < ''z'' is necessary: if it were not met, then the three numbers ''a'' = ''z'', ''b'' = ''x'' + ''y'', and ''c'' = ''x'' − ''y'' would still satisfy the equation 2''a''
2 = ''b''
2+ ''c''
2 characterizing automedian triangles, but they would not satisfy the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
and could not be used to form the sides of a triangle.
Consequently, using
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
that generates primitive
Pythagorean triangle
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s it is possible to generate primitive ''integer'' automedian triangles (i.e., with the sides sharing no common factor) as
:
:
:
with
and
coprime,
odd, and to satisfy the triangle inequality