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Inscribed and Circumscribed Circles

Introduction

Inscribed and circumscribed circles are special circles related to triangles. An incircle is a circle drawn inside a triangle that touches all three sides, while a circumcircle is a circle drawn outside the triangle that passes through all three vertices.

To construct these circles, we need to find special points called the incentre (for the incircle) and circumcentre (for the circumcircle). These points are found using angle bisectors and perpendicular bisectors respectively.

Finding the Incentre

Steps to Find the Incentre

  1. Construct triangle ABC using a ruler and compass under given conditions
  2. Bisect at least two angles of the triangle (angle A and angle C)
  3. Extend the angle bisectors until they intersect at point L (the incentre)
  4. Measure the perpendicular distance from L to each side (AB, AC, BC)

Observation:

The distances from the incentre to all three sides of the triangle are equal.

This equal distance is the radius of the incircle.

Drawing the Incircle

Steps to Draw the Incircle

  1. After finding the incentre (L) as described above
  2. Measure the perpendicular distance from L to any side (this is the radius)
  3. Place your compass at point L and draw a circle with the measured radius
  4. The circle should touch all three sides of the triangle

Note:

The incircle is the largest circle that fits inside the triangle and touches all three sides.

Finding the Circumcentre

Steps to Find the Circumcentre

  1. Construct triangle ABC using a ruler and compass under given conditions
  2. Find the perpendicular bisector of at least two sides (AB and BC)
  3. Extend the perpendicular bisectors until they intersect at point S (the circumcentre)
  4. Measure the distance from S to each vertex (A, B, C)

Observation:

The distances from the circumcentre to all three vertices of the triangle are equal.

This equal distance is the radius of the circumcircle.

Drawing the Circumcircle

Steps to Draw the Circumcircle

  1. After finding the circumcentre (S) as described above
  2. Measure the distance from S to any vertex (this is the radius)
  3. Place your compass at point S and draw a circle with the measured radius
  4. The circle should pass through all three vertices of the triangle

Note:

The circumcircle is the smallest circle that fits around the triangle and passes through all three vertices.

Practice Exercise

Question 1

Construct an equilateral triangle ABC with sides 6cm. Find its incentre and draw its incircle.

Steps:

  1. Draw triangle ABC with AB = BC = CA = 6cm
  2. Bisect angles A and B to find the incentre
  3. Measure the perpendicular distance from incentre to any side
  4. Draw the incircle with this radius

Question 2

Construct a right-angled triangle ABC with AB = 4cm, BC = 3cm and angle B = 90°. Find its circumcentre and draw its circumcircle.

Steps:

  1. Draw triangle ABC with right angle at B
  2. Find perpendicular bisectors of AB and BC
  3. Their intersection is the circumcentre
  4. Measure distance to any vertex and draw the circumcircle

Question 3

Construct a scalene triangle ABC with AB = 5cm, BC = 6cm and AC = 7cm. Find both its incentre and circumcentre.

Steps:

  1. Draw triangle ABC with given sides
  2. For incentre: Bisect angles A and C
  3. For circumcentre: Find perpendicular bisectors of AB and BC
  4. Note their different positions

Question 4

Construct an isosceles triangle ABC with AB = AC = 5cm and BC = 6cm. Draw both its incircle and circumcircle.

Steps:

  1. Draw triangle ABC with AB = AC = 5cm, BC = 6cm
  2. Find incentre by bisecting angles A and B
  3. Find circumcentre by bisecting sides AB and BC
  4. Draw both circles with appropriate radii

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