Construction of Circumcircle and Incircle: Circle is the loci of all the points at the same distance from a fixed point. A circumcircle is a circle drawn to circumscribe the given polygon. An incircle is a circle drawn inside the polygon that touches all sides of the polygon. The circumcircle is a circle that passes through all the vertices of a triangle (polygon). The centre of the circle is the point of intersection of the perpendicular bisectors of the sides of the triangle (polygon).

The incircle is a circle inscribed in the triangle (polygon), and the centre of the circle is the point of intersection of the angular bisectors of the triangle (polygon). In this article, we are going to study the constructions of the circumcircle, incircle of the triangle.

Circumcircle

The circle that circumscribes the triangle or polygon is called the circumcircle. The circumcircle of the triangle passes through all three vertices of the triangle. The centre of the circumcircle is called the circumcentre. 

The circumcentre of a circle is the point of intersection of the perpendicular bisectors of the sides of the triangle. The circumcentre of the circle can be denoted by the letter \(S.\) The distance of the circumcentre to the vertex of the triangle is known as the circumradius of the circle.

Learn About Incircle of a Triangle

Incircle

The incircle is a circle inscribed in the triangle (polygons). The incircle is a circle that passes through the mid-points of all three sides of the triangle. The centre of the incircle is called the incentre. The incentre of the circle is a point of intersection of the internal angle bisectors of the triangle. \(“I”\) denotes the incentre of the circle. 

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Construction of Perpendicular Bisector

The perpendicular bisector of a line is the line drawn at the right angles to the given line, dividing the given line into two equal parts. The steps of construction of a perpendicular bisector to the given line are listed below:

Step-1: Draw a line \(AB\) of any length by using the ruler.

Step-2: By taking the compass of radius equals to more than half of the length of the given line and placing the tip of the compass at one point of the line (say \(A)\), draw an arc above and below the given line.

Step-3: Similarly, from the other point (say \(B)\), draw another arc that intersects the previous arc by using the compass of the same radius above and below the given line.

Step-4: Join the intersection points of arcs \(C\) and \(D\) and extend the line drawn, which gives the perpendicular bisector to the given line.

Step-5: The line \(CD\) thus obtained is the perpendicular bisector of the line segment \(AB. CD\) will bisect \(AB\) at point \(M.\)

Construction of Angle Bisector

An angle bisector is a line drawn from the centre of the angle that bisects the given angle. The steps of construction of the angle bisector of the angle are given below:

i. At a point \(O,\) construct an angle of a given measurement by using the protractor.

ii. From point \(O,\) draw an arc of any radius to cut the angle’s sides at points \(A\) and \(B.\)

iii. Draw an arc of radius more than half of \(AB\) by placing the tip of the compass at point \(A\) of the interior of the angle.

iv. Draw another arc of the sam radius by placing the tip of the compass at point \(B\) of the interior of the angle that intersects the previous arc.

v. Name the intersection point as \(C.\) Join \(O\) and \(C.\)

Thus the ray drawn from the points \(O\) and \(C\) gives the angle bisector of the given angle.

Construction of Circumcircle

The circumcircle is a circle that circumscribes the triangle. To construct the circumcircle, we need a circumcentre. The circumcentre is the point of concurrency of the perpendicular bisectors of the sides. The steps of construction of the circumcircle that passes through the vertices of the triangle are discussed below:

i. First, draw a triangle (say \(ABC)\) of given measurements.

ii. Construct the perpendicular bisector to any one side of the triangle (Say \(AB)\).
Place the tip of the protractor at point \(A\) and draw arcs above and below the side \(AB\) by taking the radius more than half of the side \(AB\). Again, draw arcs from point \(B,\) and draw arcs above and below the line \(AB\) using the compass of the same radius. Join the intersection points of the arcs drawn.

iii. Next, construct the perpendicular bisector to the side of the triangle (say \(BC).\)

iv. Place the tip of the protractor at point \(B\) and draw arcs above and below the side \(BC\) by taking the radius more than half of the side \(BC.\) Again, draw arcs from point \(C,\) and draw arcs above and below the line \(BC\) using the compass of the same radius. Join the intersection points of the arcs drawn.

v. The point of intersection of the perpendicular bisectors of the sides \(AB\) and \(AC\) drawn is the circumcentre \(O.\)

vi. By placing the tip of the compass at point \(O\) and taking the radius \((OA=OB=OC),\) draw a circle.

Thus, the circle so formed through the points \(A, B, C\), gives the circumcircle.

Construction of Incircle

The incircle is the circle inscribed in the triangle. To construct the incircle, we need to construct the incentre, which is the point of concurrency of the angle bisectors of the triangle. The steps of construction of incircle are given below:

i. First, draw a triangle (say \(ABC)\) of the given measurement.

ii. Now, construct the angle bisector of any angle (say \(A)\) of the triangle \(ABC.\)

Draw arcs by placing the tip of the compass at point \(A\) by using any radius that cuts the sides at \(P\) and \(Q.\) Now, again, draw arcs inside the triangle by placing the compass at the points \(P\) and \(Q\) of more than half of the \(PQ.\)

iii. Again construct the angle bisector of another angle (say \(B).\)

Draw arcs by placing the tip of the compass at point \(B\) by using any radius that cuts the sides at \(R\) and \(S.\) Now, again, draw arcs inside the triangle by placing the compass at the points \(R\) and \(S\) of more than half of the \(RS.\)

iv. Now the intersection point of the angle bisectors of the angles \(A\) and \(B\) is the incentre \(I\) of the triangle.

v. Now construct a perpendicular from the point \(I\) on any side (say \(AC)\) to intersects the side \(AC\) at \(M.\)

vi. By placing the tip of the compass at \(I\) and taking the radius \((IM),\) draw a circle, which is the incircle of the given triangle.

Solved Examples – Construction of Circumcircle and Incircle

Q.1. Construct the incircle of the triangle ABC as given below:

Ans: In the given triangle \(ABC,AB = 7\;{\rm{cm}},BC = 6\;{\rm{cm}},\), and \(A B C, A B=7 \mathrm{~cm}, B C=6 \mathrm{~cm}\)
i. Now, construct the angle bisector of any angle (say \(A)\) of the triangle \(ABC.\)
ii. Again construct the angle bisector of another angle (say \(B).\)
iii. Now the intersection point of the angle bisectors of the angles \(A\) and \(B\) is the incentre \(I\) of the triangle.

iv. Now draw a perpendicular to any side (say \(AB)\) to intersects the side \(AB\) at \(D.\)

v. Now draw the circle with the centre \(I\) and taking the radius \(ID,\) which gives the incircle of the triangle \(ABC.\)

Q.2. Construct the circumcentre to the given triangle.

Ans:
i. Construct the perpendicular bisectors of any two sides of the triangle (say \(AC\) and \(BC).\)
ii. Let the point of intersection of these perpendicular bisectors is \(“S”.\)
iii. Now, draw a circle with centre \(“S”\) and radius \(SA=AB=SC,\) which gives the circumcircle of the given triangle.

Q.3. Construct the incircle of the equilateral triangle.

Ans:
i. Now, construct the angle bisector of any angle (say \(P)\) of the triangle \(PQR.\)
ii. Again construct the angle bisector of another angle (say \(Q).\)
iii. Now the intersection point of the angle bisectors of the angles \(P\) and \(Q\) is the incentre \(I\) of the triangle.
iv. Now draw a perpendicular from the point \(I\) on any side (say \(PQ)\) to intersects the side \(PQ\) at \(S.\)
v. Now draw the circle with centre \(I\) and taking the radius \(IS,\) which gives the incircle of the triangle \(PQR.\)

Q.4. Construct a circumcircle for the triangle given below:

Ans:
i. Construct the perpendicular bisectors of any two sides of the triangle (Say \(AC\) and \(BC).\)
ii. Let the point of intersection of these perpendicular bisectors is \(“O”.\)
iii. Now, draw a circle with centre \(“O”\) and radius \(OA=OB=OC,\) which gives the circumcircle of the given triangle.

Q.5. Identify the incentre in the given figure.

Ans:
The incentre of the circle is a point of intersection of the internal angle bisectors of the triangle.
Here the lines \(PM\) and \(QN\) are the angle bisectors of the angles \(P\) and \(Q\) of the triangle \(PQR\) as shown in the figure.
The intersection point of the lines \(PM\) and \(QN\) are \(B,\) which is the incentre of the given triangle.

Summary

In this article, we have discussed the definitions of incentre, circumcentre, circumcircle, and incircle. This article gives the constructions of perpendicular bisectors and angle bisectors. Here, we have discussed the constructions of the circumcircle and the incircle with the solved examples that help us understand easily.

Learn About Circumcircle of a Triangle

FAQs

Q.1. How to construct a circumcircle?
Ans:
i. Construct perpendicular bisectors of any two sides of the triangle.
ii. The intersection point of perpendicular bisectors is circumcentre.
iii. Using this circumcentre and taking the radius equals to distance between the circumcentre to the vertex of a triangle, draw a circle.
iv. This circle gives the circumcircle.

Q.2. What is incircle?
Ans: The incircle is a circle inscribed in the triangle (polygons). The incircle is a circle that touches all three sides of the triangle.

Q.3. What is the circumcentre of the triangle?
Ans: The circumcentre of the circle is the point of intersection of the perpendicular bisectors of the sides of the triangle.

Q.4. How do you construct an incircle to the given triangle?
Ans:
i. Construct angular bisectors of any two angles of the triangle.
ii. The intersection point of these lines is incentre.
iii. Draw any perpendicular from incentre on any one side.
iv. By taking the incentre as a centre and taking the radius equals the perpendicular distance of the incentre to a side, draw a circle.
v. Thus, a circle so formed is incircle.

Q.5. What is the incentre of a circle?
Ans: The incentre of the circle is a point of intersection of the internal angle bisectors of the triangle.

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