If a, b and c are the sides of a triangle and $$a^2+b^2+c^2=ab+bc+ca$$​, then the triangle is:

Correct option is A

Given:

The sides of a triangle are a, b, and c, and it is given that:

​$$a^2 + b^2 + c^2 = ab + bc + ca$$​

Solution:

We start by rearranging the equation:

​$$a^2 + b^2 + c^2 - ab - bc - ca = 0$$​

We can factor this expression using a known identity. This expression simplifies to:

​$$\frac{1}{2} \left[ (a - b)^2 + (b - c)^2 + (c - a)^2 \right] = 0$$​

For this sum of squares to be zero, each individual square term must be zero. Hence, we have:

​$$(a - b)^2 = 0, \quad (b - c)^2 = 0, \quad (c - a)^2 = 0$$​

This implies:

a = b = c

Conclusion: Since all the sides a, b, and c are equal, the triangle is an equilateral triangle.