# The Eutrigon Theorem

Eutrigon is a triangle (in general scalene), with one angle equal $60^{\circ}$. The side opposite the $60^{\circ}$ angle is called (unfortunately in my view) the hypotenuse, the other two sides are called the legs.

The area of any eutrigon is equal to the sum of the areas of the equilateral triangles on its legs, minus the area of the equilateral triangle on its hypotenuse.

The applet below serves a dynamic illustration of a visual proof:

17 January 2015, Created with GeoGebra

Let the legs of an eutrigon are $a$ and $b$ and the hypotenuse $c$, $A$ its area and $A(x)$ the area of an equilateral triangle with side $x$, then what we see in the applet can be expressed by two equations:

$A(a+b)=A(a)+A(b)+2A$ and
$A(a+b)=A(c)+3A$,

from which the theorem follows.

• Equilateral Triangles on Sides of a Quadrilateral
• Euler Line Cuts Off Equilateral Triangle
• Four Incircles in Equilateral Triangle
• Problem in Equilateral Triangle
• Problem in Equilateral Triangle II
• Sum of Squares in Equilateral Triangle
• Triangle Classification
• Isoperimetric Property of Equilateral Triangles
• Maximum Area Property of Equilateral Triangles
• Angle Trisectors on Circumcircle
• Equilateral Triangles On Sides of a Parallelogram
• Pompeiu's Theorem
• Circle of Apollonius in Equilateral Triangle