The point of concurrency of three altitudes of a triangle is called its :

Question

Accepted Answer

The point of concurrency of three altitudes of a triangle is called its orthocentre. Let's delve into why Option 3 is the correct answer and why the other options are incorrect.

1. **Explanation for Option 3 (Orthocentre):**
   - In any triangle, an altitude is a perpendicular segment from a vertex to the line containing the opposite side. The orthocentre is the point where all three altitudes of a triangle intersect. This point of concurrency is unique to the altitudes and is a fundamental concept in triangle geometry. The orthocentre can lie inside the triangle for acute triangles, on the triangle for right triangles, and outside the triangle for obtuse triangles. This property is consistent across all types of triangles, making the orthocentre the definitive point of concurrency for altitudes.

2. **Explanation for why other options are incorrect:**
   - **Option 1 (Incentre):** The incentre is the point of concurrency of the angle bisectors of a triangle. It is the center of the circle that can be inscribed within the triangle (incircle). The incentre is not related to the altitudes, hence it is not the correct answer.
   - **Option 2 (Centroid):** The centroid is the point of concurrency of the medians of a triangle. A median is a line segment joining a vertex to the midpoint of the opposite side. The centroid is the center of mass or balance point of the triangle, but it is unrelated to the altitudes.
   - **Option 4 (Circumcentre):** The circumcentre is the point of concurrency of the perpendicular bisectors of the sides of a triangle. It is the center of the circle that can circumscribe the triangle (circumcircle). Like the incentre and centroid, the circumcentre is not associated with the altitudes.

In summary, the orthocentre is the correct answer because it specifically refers to the intersection of the altitudes of a triangle, distinguishing it from the other points of concurrency associated with different sets of lines within a triangle.