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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.**

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.

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.

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

LaShay built a triangular shaped farm and put gates in each corner.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.

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Consider the definitions of the circles of a triangle.

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.

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.

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.

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

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.