Automedian triangle

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 three medians of an automedian triangle may be translated 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

{\displaystyle 2a^{2}=b^{2}+c^{2}}

or a permutation thereof, analogous to the Pythagorean theorem characterizing right triangles as the triangles satisfying the formula

a

2

+

b

2

=

c

2

{\displaystyle a^{2}+b^{2}=c^{2}}

.

Equivalently, in order for the three numbers

a

{\displaystyle a}

,

b

{\displaystyle b}

, and

c

{\displaystyle c}

to be the sides of an automedian triangle, the sequence of three squared side lengths

b

2

{\displaystyle b^{2}}

,

a

2

{\displaystyle a^{2}}

, and

c

2

{\displaystyle c^{2}}

should form an arithmetic progression. That is,

b

2

−

k

=

a

2

{\displaystyle b^{2}-k=a^{2}}

, and

a

2

−

k

=

c

2

{\displaystyle a^{2}-k=c^{2}}

(for example, if

b

=

17

{\displaystyle b=17}

,

a

=

13

{\displaystyle a=13}

, and

c

=

7

{\displaystyle c=7}

, then

k

=

120

{\displaystyle k=120}

:

17

2

−

120

=

13

2

=

169

{\displaystyle 17^{2}-120=13^{2}=169}

, and

13

2

−

120

=

7

2

=

49

{\displaystyle 13^{2}-120=7^{2}=49}

).

Construction from right triangles

If

x

{\displaystyle x}

,

y

{\displaystyle y}

, and

z

{\displaystyle z}

are the three sides of a right triangle, sorted in increasing order by size, and if

2

x

<

z

{\displaystyle 2x<z}

, then

z

{\displaystyle z}

,

x

+

y

{\displaystyle x+y}

, and

y

−

x

{\displaystyle 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

{\displaystyle 2x<z}

is necessary: if it were not met, then the three numbers

a

=

z

{\displaystyle a=z}

,

b

=

x

+

y

{\displaystyle b=x+y}

, and

c

=

y

−

x

{\displaystyle c=y-x}

would still satisfy the equation

2

a

2

=

b

2

+

c

2

{\displaystyle 2a^{2}=b^{2}+c^{2}}

characterizing automedian triangles, but they would not satisfy the triangle inequality and could not be used to form the sides of a triangle.

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

a

=

m

2

+

n

2

b

=

m

2

+

2

m

n

−

n

2

c

=

|

m

2

−

2

m

n

−

n

2

|

{\displaystyle {\begin{aligned}a&=m^{2}+n^{2}\\b&=m^{2}+2mn-n^{2}\\c&=|m^{2}-2mn-n^{2}|\\\end{aligned}}}

with

m

{\displaystyle m}

and

n

{\displaystyle n}

coprime,

m

+

n

{\displaystyle m+n}

odd, and to satisfy the triangle inequality

n

<

m

<

n

3

{\displaystyle n<m<n{\sqrt {3}}}

(if the quantity inside the absolute value signs is negative) or

m

>

(

2

+

3

)

n

{\displaystyle m>(2+{\sqrt {3}})n}

(if that quantity is positive). Then this triangle's medians

t

a

,

t

b

,

t

c

{\displaystyle t_{a},t_{b},t_{c}}

are found by using the above expressions for its sides in the general formula for medians:

t

a

=

2

b

2

+

2

c

2

−

a

2

4

=

a

3

2

,

t

b

=

2

a

2

+

2

c

2

−

b

2

4

=

c

3

2

,

t

c

=

2

a

2

+

2

b

2

−

c

2

4

=

b

3

2

,

{\displaystyle t_{a}={\sqrt {\frac {2b^{2}+2c^{2}-a^{2}}{4}}}={\frac {a{\sqrt {3}}}{2}},\qquad t_{b}={\sqrt {\frac {2a^{2}+2c^{2}-b^{2}}{4}}}={\frac {c{\sqrt {3}}}{2}},\qquad t_{c}={\sqrt {\frac {2a^{2}+2b^{2}-c^{2}}{4}}}={\frac {b{\sqrt {3}}}{2}},}

where the second equation in each case reflects the automedian feature

2

a

2

=

b

2

+

c

2

.