A point P lies inside a triangle and is equidistant from all three vertices of the triangle. Which special point of the triangle does P represent?

Introduction / Context:
This question is about identifying special points associated with a triangle: centroid, incenter, orthocentre, circumcentre and excentre. The key information is that the point inside the triangle is equidistant from all three vertices. The problem tests whether you can correctly match this geometric property with the name of the corresponding centre, which is a crucial concept in Euclidean geometry and many competitive exams.

Given Data / Assumptions:

- P is a point located inside a given triangle.
- Distances from P to the vertices A, B and C are equal, that is PA = PB = PC.
- No information is provided about distances from P to the sides or about right angles or medians.
- We must determine which triangle centre P represents among centroid, incenter, orthocentre, circumcentre and excentre.

Concept / Approach:
Each special centre of a triangle has a characteristic definition:
- The centroid is the intersection of medians and is the centre of mass of the triangle.
- The incenter is equidistant from all three sides and is the centre of the inscribed circle.
- The orthocentre is the intersection of altitudes and is not defined by equal distances to vertices or sides.
- The circumcentre is the intersection of perpendicular bisectors of the sides and is equidistant from all three vertices.
- Excentres are centres of external circles tangent to one side and extensions of the other two sides and are not equidistant from all vertices. The only centre that is equidistant from all vertices is the circumcentre.

Step-by-Step Solution:
Step 1: Recall that the circumcentre is defined as the point that is equidistant from all three vertices of the triangle and is the centre of the circumcircle.Step 2: The problem states that P is equidistant from all three vertices, so PA = PB = PC.Step 3: This exactly matches the definition of the circumcentre.Step 4: The centroid, incenter and orthocentre do not have this property of equal vertex distances.Step 5: The incenter is equidistant from sides, not vertices.Step 6: The orthocentre is defined by altitudes and their intersection, not by any distance equality.Step 7: Therefore, the correct identification of P is the circumcentre.

Verification / Alternative check:
Visualise a triangle and its circumcircle. The circumcentre is the centre of that circle and lies at the intersection of perpendicular bisectors of the sides. Since all vertices lie on the circumcircle, the distance from the circumcentre to each vertex is the same, equal to the circumradius. No other standard centre maintains equal distances to all three vertices. This confirms that any point equidistant from the vertices must be the circumcentre.

Why Other Options Are Wrong:
The centroid is the average of the vertex positions and lies two thirds of the way along each median, but its distances to the vertices are generally different. The incenter is equidistant from all three sides, not from the vertices. The orthocentre can lie inside, outside or on the triangle, depending on the type of triangle, and has no symmetry in distances to vertices. Excentres lie outside the triangle and are defined relative to sides and their extensions, not equal vertex distances. Therefore none of these match the condition given.

Common Pitfalls:
Students often confuse circumcentre and incenter because both relate to circles associated with the triangle. Remember the guiding keyword: circumcentre means circle passing through the vertices, so equal distances to vertices; incenter means circle touching the sides, so equal distances to sides. Mistaking one for the other is a frequent error in multiple choice questions like this.

Final Answer:
Hence, the point P is the Circumcentre of the triangle.