Circumcentre Formula

Circumcentre of a triangle is a special point in the triangle at which the perpendicular bisectors of all three sides bisect. In simple words, the point of bisection of perpendicular bisectors of the sides of a triangle is what we call the circumcentre of a triangle. This is also the centre of the circle, crossing the vertices of the given triangle. That is to say, Circumcentre is also at an equal distance to all the vertices of the triangle. The triangle’s circumcentre lies inside for an acute-angled triangle, outside for an obtuse-angled triangle and at the centre of the hypotenuse for a right-angled triangle.

Perpendicular Bisector and Circumcentre of a Triangle

For better clarity and enhanced understanding of the circumcentre of a triangle, we will require to properly understand the concept of perpendicular bisector.

If we draw a perpendicular bisector of a line segment MN, then the arbitrary point P on the perpendicular bisector will be at an equal distance from the end points M, N of that line segment.

Method to Calculate The Circumcentre of a Triangle

1. Determine the midpoint of the sides provided the coordinates MN NO OM.

2. Find out the slope of a specific line.

3. By applying the slope and midpoint find out the equation of the line segment y - y₁ = m[x - x₁].

4. Calculate the equation of the other line in a similar manner.

5. Solve the 2 bisector equations by computing the interception point calculated interception point is the Circumcentre.

6. Circumcentre can be computed using the linear equations.

Properties of the Circumcentre

Applying properties of the circumcentre, we can even simply solve problems that themselves don't involve the circumcentre directly. Because a circumcentre is a substantial structure that correlates lengths and angles, using it appropriately in a problem can be very yielding. For this reason, it is significant to know how to locate circumcentres and a proper moment to use them.

O is the circumcentre of PQR if and only if:

1. PO = QO = RO

2. QO = RO and angle ∠ POR = 2 ∠P is acute and PO are on the same side of QR.

3. QO = RO and angle ∠ POR = 2 (180 - ∠ P) when ∠P is obtuse and P, O are on opposite sides of QR.

Centroid of a Triangle

Before getting to learn the mathematical term centroid, let’s first know what we refer to by a median. In that context, a median is the line connecting the mid-points of the sides and the opposite vertices. A centroid is typically the point of bisection of the medians of the triangle and also divides the median in the ratio of 2:1.

Let’s understand the concept of the centroid of a triangle using a solved example.

Example: Identify the centroid of the triangle the coordinates of whose vertices are assigned as M(x₁, y₁), N(x₂, y₂) and O(x₃, y₃) respectively.

Solution: Now, we already know that a centroid divides the median of a triangle in the ratio 2:1. Therefore, since ‘G’ is the median, thus MG/AQ = 2/1.

Seeing that, D is tG is the Centroid of a triangle the midpoint of NO, coordinates of D are as follows:

((x₂ + x₃)/2, (y₂ + y₃)/2)

Applying the section formula, the coordinates of G will be;

{2(x₂ + x₃)/2} + 1.x₁/2 + 1, {2(y₂ + y₃)/2} + 1.y₁/2 +1]

⇒ Thus, Coordinates of G are [x₁ + x₂ + x₃/3, y₁ + y₂ + y₃/3]

If the coordinates of a triangle are (x₁, y₁), (x₂, y₂) and (x₃, y₃), then the coordinates of the centroid (which is usually represented by G) are given in the following manner:

[x₁ + x₂ + x₃/3, y₁ + y₂ + y₃/3].

Incentre of a Triangle

The incentre of a triangle is the point of bisection of the angle bisectors of angles of the triangle. An incentre is also referred to as the centre of the circle that touches all the sides of the triangle.

The Angle bisector typically splits the opposite sides in the ratio of remaining sides i.e. MP/PO = MN/MO = o/n.

Incentre splits the angle bisectors in the stated ratio of (n + o):a, (o + m):n and (m + n):o.