Name the type of triangle in which the orthocentre, circumcentre, incentre and centroid all coincide at a single point.

Introduction / Context:
This conceptual geometry question asks about special points associated with a triangle, namely the orthocentre, circumcentre, incentre and centroid. These points are defined through different constructions, and their relative positions tell us important information about the type of triangle.

Given Data / Assumptions:

• We have a triangle in which four special centres are considered.
• Orthocentre: intersection of the three altitudes.
• Circumcentre: intersection of the perpendicular bisectors of the sides.
• Incentre: intersection of the angle bisectors.
• Centroid: intersection of the medians.
• We are told all four coincide at a single point.

Concept / Approach:
In general triangles, these four centres are distinct. However, for certain special triangles, some or all of these centres can coincide. The key fact is:
In an equilateral triangle, all four centres (orthocentre, circumcentre, incentre and centroid) are at the same point due to the perfect symmetry of all sides and angles.
No other triangle type shares this complete coincidence of all four centres.

Step-by-Step Solution:
Step 1: Recall that an equilateral triangle has all sides equal and all angles equal to 60 degrees. Step 2: Because of this high degree of symmetry, the perpendicular bisectors, altitudes, angle bisectors and medians all lie on the same three lines. Step 3: The intersection point of these lines is simultaneously the orthocentre, circumcentre, incentre and centroid. Step 4: For other triangle types, these centres do not coincide completely. For instance, in a right angled triangle, the circumcentre lies at the midpoint of the hypotenuse, while the orthocentre is at the right angle vertex. Step 5: Therefore, the only triangle in which all four centres coincide is an equilateral triangle.

Verification / Alternative check:
Consider coordinate geometry. An equilateral triangle placed symmetrically around the origin will have all its symmetry lines intersecting at the origin. Using formulas for centroid and circumcentre in symmetric placement shows they are identical, and by definition the incentre and orthocentre also lie at that same point because of equal angles and equal side lengths.

Why Other Options Are Wrong:
Isosceles triangle: While some centres may lie on the same vertical line, they usually do not coincide at a single point unless the triangle is equilateral.
Right angled triangle: The orthocentre is at the right angle vertex, but the circumcentre is at the midpoint of the hypotenuse; they are not the same point.
Obtuse angled triangle: The circumcentre lies outside the triangle, while the orthocentre lies inside or at a vertex, so they do not coincide.
Scalene triangle: There is no symmetry at all; all four centres are completely distinct.

Common Pitfalls:
Many learners confuse isosceles and equilateral triangles, thinking that equal sides alone are enough for all centres to coincide. However, equal sides in an isosceles triangle still leave one angle different. Only in an equilateral triangle are all sides and all angles equal, giving full symmetry and coinciding centres.

Final Answer:
The triangle is an equilateral triangle.