Prove that the triangle is equilateral if ,2a^2+b^2+c^2=2a(b+c), when a,b,c are the lenghts of the sides of a triangle.

If a,b,c, are the lengths of the sides of an equilateral
triangle, that means that:

a = b =
c

So, one way to solve the problem is to substitute b and c
by a, in the given relation:

2a^2 + a^2 + a^2 =
2a(a+a)

We'll combine like terms and we'll
get:

4a^2 =
2a*2a

4a^2 = 4a^2
q.e.d.

Another method of solving the proble
is to remove the brackets from the right
side:

2a^2+b^2+c^2=2a(b+c)

2a^2+b^2+c^2
= 2ab + 2ac

We'll subtract both sides 2ab +
2ac:

2a^2+b^2+c^2 - 2ab - 2ac =
0

We'll write 2a^2 = a^2 +
a^2

We'll combine the terms in such way to complete the
squares:

(a^2 - 2ab + b^2) + (a^2 - 2ac + c^2) =
0

(a-b)^2 + (a-c)^2 = 0

We'll
impose the constraint of an equilateral triangle, that:

a =
b = c

and we'll substitute b by a and c by
a:

(a-a)^2 + (a-a)^2 = 0

0 + 0
= 0

0 = 0
q.e.d.

So, the relation holds
if and only if the triangle is equilateral.