Centroid, Incentre, Circumcentre, Orthocentre in a Right Angle and Equilateral Triangle

In geometry, the centroid, incentre, circumcentre, and orthocentre are four significant points associated with triangles. Each of these points has unique properties and plays a crucial role in triangle geometry. Let's explore these points in the context of right-angled and equilateral triangles.

Centroid

The centroid (G) of a triangle is the point where the three medians of the triangle intersect. A median is a line segment joining a vertex to the midpoint of the opposite side. The centroid is also known as the center of gravity, as it is the balance point of the triangle.

Formula

In a triangle with vertices at coordinates $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the centroid $(G_x, G_y)$ is given by:

$$ G_x = \frac{x_1 + x_2 + x_3}{3}, \quad G_y = \frac{y_1 + y_2 + y_3}{3} $$

Example

In an equilateral triangle, the centroid coincides with the incentre, circumcentre, and orthocentre. In a right-angled triangle, the centroid lies on the median from the right angle, dividing it in a 2:1 ratio.

Incentre

The incentre (I) is the point where the angle bisectors of a triangle intersect. It is also the center of the inscribed circle (incircle), which touches all three sides of the triangle.

Formula

The coordinates of the incentre can be found using the formula:

$$ I_x = \frac{a x_1 + b x_2 + c x_3}{a + b + c}, \quad I_y = \frac{a y_1 + b y_2 + c y_3}{a + b + c} $$

where $a$, $b$, and $c$ are the lengths of the sides opposite to the vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, respectively.

Example

In an equilateral triangle, the incentre is the same as the centroid. In a right-angled triangle, the incentre lies inside the triangle and is equidistant from all sides.

Circumcentre

The circumcentre (O) is the point where the perpendicular bisectors of the sides of a triangle intersect. It is the center of the circumscribed circle (circumcircle) that passes through all three vertices of the triangle.

Formula

For a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the circumcentre can be found using the circumcircle's equation or by solving the system of equations of the perpendicular bisectors.

Example

In an equilateral triangle, the circumcentre is at the same point as the centroid. In a right-angled triangle, the circumcentre is at the midpoint of the hypotenuse.

Orthocentre

The orthocentre (H) is the point where the three altitudes of a triangle intersect. An altitude is a perpendicular line from a vertex to the opposite side (or its extension).

Formula

The orthocentre does not have a straightforward formula like the centroid or incentre, as it involves solving the equations of the altitudes of the triangle.

Example

In an equilateral triangle, the orthocentre coincides with the other three centers. In a right-angled triangle, the orthocentre is at the vertex forming the right angle.

Comparison Table

Point	Right-Angled Triangle	Equilateral Triangle	Formula
Centroid (G)	Lies on the median from the right angle	Coincides with other centers	$G_x = \frac{x_1 + x_2 + x_3}{3}$, $G_y = \frac{y_1 + y_2 + y_3}{3}$
Incentre (I)	Lies inside the triangle	Coincides with other centers	$I_x = \frac{a x_1 + b x_2 + c x_3}{a + b + c}$, $I_y = \frac{a y_1 + b y_2 + c y_3}{a + b + c}$
Circumcentre (O)	Midpoint of the hypotenuse	Coincides with other centers	Solved using perpendicular bisectors
Orthocentre (H)	At the right angle vertex	Coincides with other centers	Solved using altitudes

In summary, while the centroid, incentre, circumcentre, and orthocentre have distinct definitions and properties in general triangles, they coincide in an equilateral triangle due to its symmetry. In a right-angled triangle, these points have specific locations that relate to the triangle's sides and angles. Understanding these points is fundamental in triangle geometry and can be crucial for solving various geometric problems.