In plane geometry, an automedian triangle is a triangle in which the lengths of the three medians (the line segments connecting each vertex to the midpoint of the opposite side) are proportional to the lengths of the three sides, in a different order.
The side lengths of an automedian triangle satisfy the formula
or a permutation thereof, analogous to the Pythagorean theorem characterizing right triangles as the triangles satisfying the formula
are the three sides of a right triangle, sorted in increasing order by size, and if
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.
Consequently, using Euler's formula that generates primitive Pythagorean triangles it is possible to generate primitive integer automedian triangles (i.e., with the sides sharing no common factor) as
odd, and to satisfy the triangle inequality
(if the quantity inside the absolute value signs is negative) or
are found by using the above expressions for its sides in the general formula for medians:
where the second equation in each case reflects the automedian feature
There are 18 primitive integer automedian triangles, shown here as triples of sides
: For example, (26, 34, 14) is not a primitive automedian triple, as it is a multiple of (13, 17, 7) and does not appear above.
is the area of the automedian triangle, by Heron's formula
[3] The Euler line of an automedian triangle is perpendicular to the median to side
where the extended medians meet the circumcircle form an isosceles triangle.
This property of automedian triangles stands in contrast to the Steiner–Lehmus theorem, according to which the only triangles two of whose angle bisectors have equal length are the isosceles triangles.
, and the Euler line of the triangle is the perpendicular bisector of
[2] When generating a primitive automedian triangle from a primitive Pythagorean triple using the Euclidean parameters
As non-primitive automedian triangles are multiples of their primitives the inequalities of the sides apply to all integer automedian triangles.
Equality occurs only for trivial equilateral triangles.
This fact allows automedian triples to have sides and perimeter of prime numbers only.
Because in a primitive automedian triangle side
is the sum of two squares and equal to the hypotenuse of the generating primitive Pythagorean triple, it is divisible only by primes congruent to 1 (mod 4).
They are also the sum and difference of the legs of a primitive Pythagorean triple.
to be divisible only by primes congruent to ±1 (mod 8).
[4] The study of integer squares in arithmetic progression has a long history stretching back to Diophantus and Fibonacci; it is closely connected with congrua, which are the numbers that can be the differences of the squares in such a progression.
[1] However, the connection between this problem and automedian triangles is much more recent.
The problem of characterizing automedian triangles was posed in the late 19th century in the Educational Times (in French) by Joseph Jean Baptiste Neuberg, and solved there with the formula
[5] Apart from the trivial cases of equilateral triangles, the triangle with side lengths 17, 13, and 7 is the smallest (by area or perimeter) automedian triangle with integer side lengths.
[2] There is only one automedian right triangle, the triangle with side lengths proportional to 1, the square root of 2, and the square root of 3.