In a triangle, the lengths of the sides are a, b, and c respectively. If a^2 + b^2 + c^2 = ab + bc + ca, what type of triangle is it?

Introduction / Context:
This algebraic geometry question asks you to classify a triangle based on a relationship between the squares and products of its side lengths. Recognising the pattern and manipulating the expression helps identify whether the triangle is equilateral, isosceles, or some other type.

Given Data / Assumptions:

• A triangle has side lengths a, b, and c.
• They satisfy the equation a^2 + b^2 + c^2 = ab + bc + ca.
• We need to determine the type of triangle (isosceles, equilateral, scalene, or right angled).

Concept / Approach:
We rearrange the given equation to see if it can be factored into a sum of squares. A well known identity states that:
(a − b)^2 + (b − c)^2 + (c − a)^2 = 2(a^2 + b^2 + c^2 − ab − bc − ca).
This shows that if a^2 + b^2 + c^2 − ab − bc − ca = 0, then the sum of these squared differences must be zero, which in turn forces a = b = c. Thus the triangle must be equilateral, with all sides equal.

Step-by-Step Solution:
Step 1: Start with the given equation: a^2 + b^2 + c^2 = ab + bc + ca. Step 2: Subtract the right side from the left side: a^2 + b^2 + c^2 − ab − bc − ca = 0. Step 3: Use the identity: (a − b)^2 + (b − c)^2 + (c − a)^2 = 2(a^2 + b^2 + c^2 − ab − bc − ca). Step 4: Substitute the expression from Step 2 into the identity: (a − b)^2 + (b − c)^2 + (c − a)^2 = 2 × 0 = 0. Step 5: A sum of squares equals zero only if each squared term is zero. Thus (a − b)^2 = 0, (b − c)^2 = 0, and (c − a)^2 = 0. Step 6: Therefore a − b = 0, b − c = 0, and c − a = 0, which implies a = b = c. Step 7: A triangle with all three sides equal is an equilateral triangle.

Verification / Alternative Check:
We can check with a specific example. Suppose a = b = c = k. Then a^2 + b^2 + c^2 = 3k^2 and ab + bc + ca = 3k^2, so the equality holds. If we try a simple non equilateral triangle, for example a = 2, b = 3, c = 4, then a^2 + b^2 + c^2 = 4 + 9 + 16 = 29 and ab + bc + ca = 6 + 12 + 8 = 26. These are not equal, so the condition does not hold. This supports the conclusion that only equilateral triangles satisfy the given relation.

Why Other Options Are Wrong:
An isosceles triangle would have at least two equal sides but not necessarily all three, and the identity above shows that all differences must vanish, not just one. A scalene triangle has all sides different, which contradicts the result that a = b = c. A right angled triangle satisfies the Pythagorean relation a^2 + b^2 = c^2 for appropriate labelling, which is a different condition and does not force a^2 + b^2 + c^2 = ab + bc + ca.

Common Pitfalls:
Some learners try to interpret the relation as a variant of the Pythagorean theorem or attempt to plug in random values instead of working with identities. Others may not recall the sum of squares identity and thus miss the elegant path to showing that all sides must be equal. Remembering and applying algebraic identities can simplify many geometry problems involving side lengths and classification of triangles.

Final Answer:
The triangle is equilateral.