Converse
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Age
14 to 16
Challenge level
Problem
Clearly if $a$, $b$ and $c$ are the lengths of the sides of a triangle and the triangle is equilateral then
$a^2 + b^2 + c^2 = ab + bc + ca$.
Is the converse true, and if so can you prove it? That is if $a^2 + b^2 + c^2 = ab + bc + ca$ is the triangle with side lengths $a$, $b$ and $c$ necessarily equilateral?
Student Solutions
Clearly if $a$, $b$ and $c$ are the lengths of the sides of a triangle and the triangle is equilateral then
Is the converse true, and if so can you prove it? That is if $a^2 + b^2 + c^2 = ab + bc + ca$ is the triangle with side lengths $a$, $b$ and $c$ necessarily equilateral?
Again you don't require much mathematical knowledge to do this, just the ability to use elementary algebra. Here is a very neat solution from Koopa Koo, Boston College, USA.
implies that
which implies $a = b = c$. So, the converse is also true.
Another, rather clumsier, method is to consider the expression
as a quadratic equation for $a$ in terms of $b$ and $c$, namely:
and then the condition for this quadratic equation to have real roots requires that $a = b = c.$