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In this lesson, the inscribed and circumscribed circles of a triangle will be constructed.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Investigating Inscribed and Circumscribed Circles of a Triangle

In a math exam, Ramsha has been given a triangle and asked to draw two circles. For one of the circles, each side of the triangle must be tangent to the circle. The second circle must pass through the three vertices of the triangle.

Triangle
Is it possible for Ramsha to draw such circles?

Discussion

Definition of Incenter, Circumcenter, and Centroid

The definition of three noticeable points will be explored below.

Recall that by the Incenter Theorem, the incenter of a triangle is equidistant from each side of the triangle.

The circumcenter is equidistant from the vertices of the triangle by the Circumcenter Theorem.

Discussion

Inscribing a Circle in a Triangle

It has been previously seen that the incenter of a triangle is equidistant from its sides. Thereofre, a circle inscribed in the triangle and centered at the incenter can be drawn.

Discussion

Circumscribing a Circle of a Triangle

To follow another connection between circles and triangles, consider the circumcenter of a triangle. Recall that this point is equidistant from the vertices of the triangle. Therefore, a circle circumscribed at the triangle and centered at the circumcenter can be drawn.

Example

Inscribed and Circumscribed Circles in Real Life

Inscribed and circumscribed circles can also be considered in real life!

LaShay built a triangular shaped farm and put gates in each corner.
Farm With Street Lamp

At night, she wants to monitor all three gates. Therefore, she will place a lamp post in her farm. Where should she place the lamp so that each of the three corners are illuminated? Define the region illuminated by the light.

Hint

Consider the definitions of the circles of a triangle.

Solution

Since LaShay wants to monitor all three gates of her farm, the lamp post must be equidistant from each corner. Recall that the circumcenter of a triangle is equidistant from its three vertices. Therefore, LaShay should place the lamp post at the circumcenter of the triangle.
Farm With Street Lamp
Note that the region region illuminated by the light makes a circle that passes through the three vertices of the triangle. Therefore, the region is the circumscribed circle of the triangle.

Closure

Summarizing Inscribed and Circumscribed Circles of a Triangle

With the topics seen in this lesson, the challenge presented at the beginning can be answered. The circle that is tangent to each side of the triangle is the inscribed circle of the triangle. The circle that passes through the three vertices of the triangle is the circumscribed circle of the triangle.

Inscribed and Circumscribed Circles
Since this lesson's focus was circles of triangles, the centroid of a triangle has not been mentioned as much as the incenter and circumcenter. However, it is important to say that the centroid of a triangle is also called the center of mass of the triangle.
Centroid

For example, consider a carpenter designing a triangular table with one leg. To determine the location of the leg, he will use the centroid of the table. Since the centroid is the center of mass, the table will be perfectly balanced.

Table with Centroid