The circumcenter of a Triangle

Circumcenter:

The circumcenter can be defined as the center of the circle of a polygon. The circumcircle of the polygon is the circle that goes across all the polygon's vertices, and the circumcenter is called the center of that circle. The circumcircle polygons are all known as cyclic polygons. However, a circumcircle doesn't need to be present in all polygons. The circumcircle and the circumcenter can only be in regular polygons, triangles, squares, and right-kites.

The Circumcenter of a Triangle

The center point of the circumcircle that encloses a triangle is known as the circumcenter of a Triangle. The circumcenter of a triangle may lie inside or outside the triangle. It depends on the type of triangle. An acute triangle will have the circumcenter inside the shape, while an obtuse triangle will contain it outside the triangle. There is another definition for the circumcenter of any triangle. The circumcenter of a triangle is situated at the intersection of the perpendicular bisectors of the sides of the triangle. This means that the point of competence of this bisector is termed the circumcenter on the sides of a triangle. It is marked with Q (A, B) in the figure given below.

The formula for the circumcenter of a triangle is quite lengthy and incorporates several properties of the triangle for its derivation.

It is given as Q (A, B) = [(a1 sin 2X + a2 sin 2Y + a3 sin 2X)/ (sin 2X + sin 2Y + sin 2X), (b1 sin 2X + b2 sin 2Y + b3 sin 2X)/ (sin 2X + sin 2Y + sin 2X)], where the coordinates of three vertices of triangle are K (a1, b1), L (a2, b2), and M (a3, b3) with X, Y, Z as the respective angles between them.

Construction of Circumcenter in a Triangle

The stepwise explanation for drawing and determining the circumcenter in any triangle are mentioned below:

Draw the required triangle for which you have to construct the circumcenter
Take a compass and draw the perpendicular bisector of each line. First, take the tip of the compass on point K and draw an arc, then place the tip on point L and draw another arc. Draw a straight line from the point of intersection of the arcs to the line KL, and this will create the perpendicular bisector. Do this for all the other lines as well.
Draw all the perpendicular bisectors and mark the point where they meet as Q.

Locating a Circumcenter

We can successfully locate a circumcenter of any triangle using two methods:

Midpoint Formula
Distance Formula

Midpoint Formula: Calculate the central points of the KL, LM, and KM segments. Calculate the slope of the KL, LM, and KM lines. Find out the equation of the perpendicular bisector line by utilizing the center point and the slope of the perpendicular line. Also, discover another bisector line equation. Solve two perpendicular bisector equations to identify the intersection. The circumcenter of the given triangle is this intersection point.

Distance Formula: The distance formula in geometry is given by d = √ (a - ax)2 + (b - bx)2. ‘d’ is the distance of a vertex from the circumcircle and x = 1, 2, 3. Calculating the values of d1 (from vertex K), d2 (from vertex L), d3 (from vertex M) and equating them together as d1 = d2 = d3, we get the coordinates of the circumcenter of the triangle KLM.