The lengths of the sides of a trlangle are `a, b and c` respectively. If `a^2 + b^2 + c^2 = ab + bc + ca`, then the triangle is:

Options

A. isosceles
B. equilateral
C. scalene
D. right-angled

Answer

To solve the problem, we need to analyze the given equation involving the sides of a triangle:

**Given:**
\( a^2 + b^2 + c^2 = ab + bc + ca \)

**Step 1: Rearranging the equation**
We can rearrange the equation as follows:
\[ a^2 + b^2 + c^2 - ab - bc - ca = 0 \]

**Step 2: Recognizing the identity**
The left side of the equation can be recognized as a specific algebraic identity. This identity can be rewritten as:
\[ \frac{1}{2} \left( (a-b)^2 + (b-c)^2 + (c-a)^2 \right) = 0 \]

**Step 3: Analyzing the identity**
For the sum of squares to equal zero, each individual square must also equal zero. Therefore, we have:
\[ (a-b)^2 = 0, \quad (b-c)^2 = 0, \quad (c-a)^2 = 0 \]

**Step 4: Concluding the equality of sides**
From these equations, we can conclude:
\[ a - b = 0 \implies a = b, \quad b - c = 0 \implies b = c, \quad c - a = 0 \implies c = a \]
Thus, we find that:
\[ a = b = c \]

**Step 5: Identifying the type of triangle**
Since all three sides of the triangle are equal, we conclude that the triangle is an equilateral triangle.

**Final Answer:**
The triangle is an equilateral triangle.

---