Construction of the Circumcircle and Incircle of a Triangle

Description

The circumcircle and incircle of a triangle are two fundamental geometric constructions. In this chapter, students will learn how to construct a circumcircle (a circle that passes through all three vertices of a triangle) and an incircle (a circle that touches all three sides of a triangle from within). This topic is explained in a structured and clear manner using practical examples and subject-specific terminology.

1. Introduction

Understanding how to construct the circumcircle and incircle of a triangle is a key skill in geometry. These constructions are not just academic but have practical applications in fields like architecture and engineering. The circumcircle is drawn around the triangle, touching all its vertices, while the incircle lies inside the triangle, touching all three sides.

2. Construction of the Circumcircle

2.1 Step 1: Drawing Perpendicular Bisectors

Begin by choosing two sides of the triangle (for example, AB and BC).
Find the midpoint of each side.
At each midpoint, draw a perpendicular bisector.
The point where these two bisectors intersect is called the circumcentre — the center of the circumcircle.

2.2 Step 2: Measure the Radius

From the circumcentre, measure the distance to any one of the triangle’s vertices (say, point A).
This distance will be the radius of the circumcircle.

2.3 Step 3: Draw the Circle

Use a compass with the radius fixed and draw a circle from the circumcentre.
This circle will pass through all three vertices of the triangle, forming the circumcircle.

Example

Let triangle ABC have sides AB = 6 cm, BC = 7 cm, and CA = 5 cm.

Find the midpoints of AB and BC.
Draw perpendicular bisectors on these sides.
Let the intersection point be O (circumcentre).
Measure OA = 6 cm.
Draw a circle with center O and radius 6 cm — this is your circumcircle.

3. Construction of the Incircle

3.1 Step 1: Draw Angle Bisectors

Draw the angle bisectors of any two angles in the triangle (for example, ∠A and ∠B).
These bisectors divide the angles into two equal parts.

3.2 Step 2: Locate the Incentre

The point where the two angle bisectors intersect is the incentre of the triangle.
The incentre is equidistant from all three sides.

3.3 Step 3: Measure the Radius

From the incentre, draw a perpendicular line to any one side of the triangle (e.g., BC).
The length of this perpendicular line is the radius of the incircle.

3.4 Step 4: Draw the Circle

Using a compass, draw a circle with the incentre as the center and the measured radius.
This circle will touch all three sides of the triangle from inside.

Example

In triangle ABC:

Draw angle bisectors of ∠A and ∠B.
Let their intersection be point P (incentre).
From P, draw a perpendicular to BC and let its length be 2 cm.
Draw a circle with center P and radius 2 cm — this is the incircle.

4. Expanded Subsections for Better Understanding

Role of Perpendicular Bisectors

The perpendicular bisectors of a triangle’s sides intersect at a point equidistant from all vertices — the circumcentre. This point ensures that the circle formed will touch all three corners of the triangle.

Properties of Angle Bisectors

An angle bisector divides an angle into two equal parts. The point of intersection of two angle bisectors is the incentre, which is equidistant from the sides of the triangle and allows for the perfect construction of the incircle.

Importance of Radius and Points of Contact

For the circumcircle, the radius connects the center to the triangle’s vertices. For the incircle, the radius is the perpendicular distance from the center to a side. These radii are essential for accurate construction and later application in solving problems.

5. Summary

Constructing the circumcircle and incircle of a triangle requires understanding and applying geometric principles.

The circumcircle is drawn by finding the intersection of two perpendicular bisectors and measuring the distance from the center to a vertex.
The incircle is created by intersecting two angle bisectors and drawing a perpendicular from the center to a side to find the radius.