Circumcentre of a triangle

Introduction

The perpendicular bisectors pass through the midpoint of the edge and form a right angle (90°) at mid-point. Now, as the triangle contains three edges and is cyclic in nature, it can easily circumscribe a circle, which leads to a circumcentre of a triangle. It is not necessary to draw all the perpendiculars to find the circumcentre of a triangle; even with two perpendicular bisectors, the circumcentre can be estimated. Circumcentre is basically a centre of the circle passing through any three points circumscribing a regular polygon like triangle, quadrilateral, rectangle, etc., but it is not valid for all the polygons to contain a circumcentre.

Circumscribed Circles

When a circle passes through the three vertices of a  triangle, the triangle is inside the circle and the triangle touches the circle with each vertex. The central point of a circular circle is called the circumcentre. Also, whenever a circle inscribes any polygon, then the sides of the polygon are tangential to the circle. In a triangle, the centre of this circle is the circumcentre. 

Constructing circumcentre of a triangle

With the help of a compass, we can locate the circumcentre inside a triangle. Now, one end of the compass is to be placed at the vertex of the triangle and another end is to be placed on the hypotenuse of the triangle.

The steps are as follows:

• Using the compass, draw the perpendicular bisector for each side of the triangle.  

• Make sure to increase the length of the perpendicular bisector to find the common intersection point for all bisectors. This intersection point is the required circumcentre of a triangle.
• Now, one end of the compass is to be placed at the vertex of the triangle and another end is to be placed on the circumcentre of the triangle to construct a circle circumscribing the triangle. 

Properties of Circumcentre of a Triangle

There are numerous properties for a circumcentre of a triangle. Some of which are listed below:

• The distance of each vertex from the circumcentre is equal.
• Isosceles triangles can be formed by joining all the vertices to the circumcentre.
• The position of the circumcentre varies for each different type of triangle.

Position of Circumcentre for Various Types of Triangle 

• Acute Angle Triangle
  Here, the circumcentre resides inside the triangle. O represents the circumcentre in the below diagram.
• Obtuse Angle Triangle 

Here, the circumcentre resides outside the triangle. O represents the circumcentre in the below diagram.

• Right Angle Triangle 

Here, the circumcentre is present on the hypotenuse of the triangle.

Here A, B, C, and M are the vertices and 2a, 2c are the side lengths. M is the midpoint of AC and represents the circumcentre. Now, if coordinates of B are (0, 0) then coordinates of M are:

M = a2+c2 , c

• Equilateral Triangle

Here, the circumcentre is present on the Euler line, i.e. circumcentre, orthocentre, incentre, and centroid all coincide.The radius of the circumcentre can be calculated by the formula below: 

radius = side length3

Circumcentre Formula in a Triangle

• If all sides lengths (a, b, c) are known, then the radius of the circumcentre is given by:

radius = abc(a+b+c)(-a+b+c)(a-b+c)(a+b-c)

• If one of the side length (a) along with angle (∠A) opposite to it is known, then the radius of the circumcentre is given by:

radius = a2sin(A)

• If the coordinates of the triangle are (x1, y1), (x2, y2) and (x3, y3), then the coordinates of the circumcentre is given by:

x1sin(2A) + x2sin(2B) + x3sin(2C)sin(2A) + sin(2B) + sin(2C), y1sin(2A) + y2sin(2B) + y3sin(2C)sin(2A) + sin(2B) + sin(2C)

• Area of circumcircle with sides a, b, and c is R2, where R is given by:

R = abc(a+b+c)(-a+b+c)(a-b+c)(a+b-c)

Incentre

The point of interaction, where all the interior angle bisectors meet, is known as the incentre of a triangle. The distance between incentre (Radius = r) and circumcentre(Radius = R) is given by: R2-2rR

Excentre

The point of interaction when the bisectors intersect,  where one of the bisectors is of interior angle and another two are of the external angle of the opposite side of the triangle.

Centroid

It is the intersection of all the three medians of the triangle (where each median passing from a vertex meets the midpoint of the opposite side).

Orthocentre

It is the point of intersection where the perpendicular drawn from the vertex to the opposite side of the triangle intersects.

Conclusion

To find a common spot between three non-collinear points, circumcentre is the position which is equidistant from all the three non-collinear points. This can be used as an application in various real-life scenarios. For example:

Let us suppose that there are 2 types of rides in an amusement park, popular and unpopular. Now, suppose the four rides in the unpopular category are very close to one another. So, what an economical park manager can do regarding the ticket selling counter is to plan a common ticket selling counter for all four rides rather than individual counters to reduce spending cost. Hence, a one-way ticket reservation box can be selected in such a way that it is close to the same distance from the four rides.