Day 126 – Inscribed and circumscribed circles of a triangle

Introduction In geometry it is possible to construct a polygon such as a triangle or a hexagon inside a circle using the basic geometrical instruments like compasses and straightedges. This implies that a circle can also be constructed on the outside of a polygon. A circle can be drawn inside a triangle such that the circle just touches all the three sides of the triangle. It is also possible to draw a circle that passes through the three vertices of the triangle. In this lesson, we will learn how to construct circles inside triangles such that the circles touch the sides of the triangle and also how to construct circles which pass through the vertices of a triangle.

Vocabulary 1. Inscribed circle (incircle) A circle which touches all the three sides of a triangle. This circle is inside the triangle. 2. Circumscribed circle (circumcircle) A circle which passes through all the vertices of a triangle. This circle is outside the triangle. 3. Incenter The center of a circle that touches all the three sides of a triangle and it is the point of intersection of the three angle bisectors of the triangle.

4. Circumcenter The center of a circumscribed circle which is the point of intersection of the perpendicular bisectors of the sides of the triangle. 5. Perpendicular bisector A line that bisects a line segment and forms a right angle at the point of intersection, which is the midpoint. 6. Angle bisector A line that bisects an angle into two angles, these angles are always congruent.

Inscribed circle of a triangle An inscribed circle is drawn inside a triangle such that the circle touches the three sides of the triangle. An inscribed circle is also referred to as an incircle. The concept of bisecting an angle using a pair of compasses is key when constructing an inscribed circle. Each triangle has its own unique incircle.

Constructing an inscribed circle of a triangle 1. Consider ∆KLM below. K L M

In order to construct an inscribed circle of ∆KLM: 2 In order to construct an inscribed circle of ∆KLM: 2. We construct the angle bisector of ∠K as shown below. K L M

3. We then construct the angle bisector of ∠L and label the point of intersection of the bisectors point O. K L M O

4. We drop a perpendicular from point O to any side of ∆KLM, in this case, side KL.

5. We label the point of intersection of the perpendicular bisector and side KL point P. M O P

6. We construct a circle with radius OP 6. We construct a circle with radius OP. This is the inscribed circle or incircle of ∆KLM. Point O is referred to as the incenter of the circle. K L M O P

Circumscribed circle of a triangle A circumscribed circle is drawn outside a triangle such that the circle passes through the three vertices of the triangle. A circumscribed circle is also referred to as a circumcircle. The concept of constructing a perpendicular bisector of a line segment using a pair of compasses is key when constructing a circumcircle. Each triangle has its own unique circumcircle.

Constructing an circumscribed circle of a triangle 1. Consider ∆MNP below. N P M

N P M In order to construct an inscribed circle of ∆MNP: 2. Construct the perpendicular bisector of side MN of ∆MNP.

3. Construct the perpendicular bisector of side NP of ∆MNP.

N P M O 4. Label the point of intersection of the two perpendicular bisectors point O.

N P M O 5. We use O as the center and use either OM, ON or OP as the radius, we draw a circle. This circle will pass through the vertices M, N and P.

We have constructed a circumscribed circle of ∆MNP We have constructed a circumscribed circle of ∆MNP. Point O is referred to as the circumcenter of the circle. Note: The circumcenter can be either be inside the triangle, outside the triangle or on the triangle. 1. It is outside the triangle when the triangle is obtuse. 2. It is inside the triangle when the triangle is acute. 3. It is on the triangle when the triangle is a right triangle.

Example Construct a circumcircle of ∆ABC below.

Solution The circumcenter will pass through the vertices A, B and C of ∆ABC. We need to perpendicular bisectors of any two sides of ∆ABC. The point of intersection of the perpendicular bisectors will be the circumcenter, O of the circle. We will use either OA, OB or OC as the radius to draw the circumcircle.

The circumcircle is constructed as shown below. P M The circumcircle is constructed as shown below.

homework Construct the incircle of the triangle below and label the incenter O.

Answers to homework The incircle is constructed as shown below. O

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