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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.

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.

Concept

Incenter

The incenter of a triangle is the point of intersection of the triangle's angle bisectors. The incenter is typically represented by the letter $I .$ This point is considered to be the center of the triangle. For every triangle, the incenter is always inside the triangle.

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

Concept

Circumcenter

The circumcenter of a triangle is the point of intersection of the triangle's perpendicular bisectors. Circumcenter of a triangle is denoted by the letter $S .$ It can be inside, outside, or on a triangle's side, depending on the triangle type.

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

Concept

Centroid

The centroid of a triangle is the point of intersection of the triangle's medians. The centroid is typically represented by the letter $G .$ This point is always inside the triangle.

Example

Identifying a Triangles Incenter, Circumcenter, and Centroid

Zain is trying to match the given points with their corresponding definitions.

Help Zain match them!

Hint

Begin by defining the segments that form the points.

Solution

To match the points with their definitions, the segments that form the points should be defined. Looking at the graph for point $A$ , it can be seen that the segments do not bisect the sides of the triangle. However, they bisect the interior angles of the triangle.

With this information, it can be concluded that the segments are the angle bisectors of the triangle. Therefore, point $A$ is the incenter of the triangle. Next, the second diagram will be considered.

The segments bisect the sides of the triangle and connect the midpoints of the sides with their opposite vertices. Therefore, they are the medians of the triangle, and point $B$ is the centroid. Lastly, the segments that form point $C$ will be defined.

Each segment is perpendicular to and bisects a side of the triangle. This means that they are perpendicular bisectors of the sides, and point $C$ is the circumcenter of the triangle.

Example

Using Incenter, Circumcenter, and Centroid in Real Life

The concepts previously investigated can also be used in real life!

In the Mile High City, Denver, the cities transportation department is planning to pave three roads that connect three neighborhoods.

Magdalena and Vincenzo are the owners of two competing hotel chains. They see this as an opportunity to expand their empires into this region. Magdalena wants her hotel to be equidistant from each paved road. On the other hand, Vincenzo wants his hotel to be equidistant from the neighborhoods.

Where should they build their hotels?

Hint

Consider the definitions of incenter and circumcenter, and centroid.

Solution

Looking at the diagram, it can be seen that the roads form a triangle. Recall that the incenter of a triangle is equidistant from each side. Also, the circumcenter of a triangle is equidistant from each vertices.

Therefore, Magdalena's hotel should be at the incenter of the triangular region. Vincenzo's hotel should be at the circumcenter of the triangular region.

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.

Concept

Inscribed Circle

The inscribed circle or incircle of a triangle is a circle that is tangent to the sides of the triangle. The center of this circle is the incenter of the triangle.

The incircle is the largest circle completely contained within a triangle.

Construction

Drawing the Inscribed Circle of a Triangle

In a few of steps, it is possible to draw the inscribed circle or incircle of a triangle.

Given a triangle, these three steps can be followed to draw its incircle.

Draw the Angle Bisectors of Two Vertices

Begin by drawing the angle bisector of two vertices of the triangle. Then, label their point of intersection as

I .

The point of intersection of the angle bisectors is the incenter of the triangle. It is also the center of the incircle.

Draw the Radius of the Incircle

The incenter is equidistant from the sides of the triangle. To draw the radius of the incircle, a line through

I

that is perpendicular to any of the sides should be drawn.

Draw the Incircle

Finally, by placing the compass at

I

and opening it along the radius, the incircle 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.

Concept

Circumscribed Circle of a Triangle

The circumscribed circle or circumcircle of a triangle is the circle that passes through the three vertices of the triangle. The center of the circumscribed circle is the circumcenter of the triangle.

Construction

Drawing the Circumscribed Circle of a Triangle

The circumscribed circle or circumcircle of a triangle can be drawn in a few of steps.

Given a triangle, these two steps can be followed to draw its circumcircle.

Draw the Perpendicular Bisectors of Two Sides

First, the perpendicular bisectors of two sides of the triangle will be drawn. Their point of intersection will be labeled as

C .

The point of intersection of the perpendicular bisectors is the circumcenter of the triangle. It is also the center of the circumcircle.

Draw the Circumcircle

The circumcircle of a triangle passes through the three vertices of the triangle. Therefore, to draw the circumcircle, a compass will be placed at point

C .

Then, the compass will be opened to match the measure between

C

and any of the vertices. Finally, with this measure, the circle 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.

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.

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.

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.

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.

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Catch-Up and Review

Hint

Solution

Hint

Solution

Hint

Solution