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3. Inscribed and Circumscribed Circles of a Triangle
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Chapter 5
3. 

Inscribed and Circumscribed Circles of a Triangle

Triangles have unique relationships with circles, particularly when it comes to inscribed and circumscribed circles. An inscribed circle is perfectly nestled inside a triangle, touching all three sides without crossing them. The center of this circle, known as the incenter, is equidistant from each side of the triangle. On the other hand, a circumscribed circle encompasses a triangle, passing through its three vertices. The center of this circle, termed the circumcenter, is equidistant from the triangle's vertices. These geometric concepts have practical applications. For instance, in urban planning, if three neighborhoods are connected by roads forming a triangle, determining the incenter can help place a facility equidistant from all three roads. Similarly, the circumcenter can help locate a facility equidistant from the neighborhoods themselves.
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9 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Inscribed and Circumscribed Circles of a Triangle
Slide of 9
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.

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 This point is considered to be the center of the triangle. For every triangle, the incenter is always inside the triangle.

Incenter

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 It can be inside, outside, or on a triangle's side, depending on the triangle type.

Circumcenter

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 This point is always inside the triangle.

Centroid
Example

Identifying a Triangles Incenter, Circumcenter, and Centroid

Zain is trying to match the given points with their corresponding definitions.
Incenter, Circumcenter, Centroid
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 , it can be seen that the segments do not bisect the sides of the triangle. However, they bisect the interior angles of the triangle.

Incenter

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

Centroid

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 is the centroid. Lastly, the segments that form point will be defined.

Circumcenter

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

Triangle Formed by Roads

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.

Incenter and Circumcenter
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.
Incenter and Circumcenter
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.
Inscribed Circle
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.

Incircle
Given a triangle, these three steps can be followed to draw its incircle.
1
Draw the Angle Bisectors of Two Vertices
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Begin by drawing the angle bisector of two vertices of the triangle. Then, label their point of intersection as
Drawing an Angle Bisector
The point of intersection of the angle bisectors is the incenter of the triangle. It is also the center of the incircle.
2
Draw the Radius of the Incircle
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The incenter is equidistant from the sides of the triangle. To draw the radius of the incircle, a line through that is perpendicular to any of the sides should be drawn.
Radius of Incircle


3
Draw the Incircle
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Finally, by placing the compass at and opening it along the radius, the incircle can be drawn.
Draw Incircle
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.

Circumscribed Circle
Construction

Drawing the Circumscribed Circle of a Triangle

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

Circumcircle
Given a triangle, these two steps can be followed to draw its circumcircle.
1
Draw the Perpendicular Bisectors of Two Sides
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First, the perpendicular bisectors of two sides of the triangle will be drawn. Their point of intersection will be labeled as
Drawing a Perpendicular Bisector
The point of intersection of the perpendicular bisectors is the circumcenter of the triangle. It is also the center of the circumcircle.
2
Draw the Circumcircle
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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 Then, the compass will be opened to match the measure between and any of the vertices. Finally, with this measure, the circle can be drawn.
Draw Circumcircle
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



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Inscribed and Circumscribed Circles of a Triangle
Exercise 2.1

In a nature reserve near the city Pitesti in Romania, there are three mysterious monuments. According to legend, these monuments are the graves of Count Dracula's three wives.

Illustration of the monuments outside Pitesti

The same legend also tells that Dracula was buried along with an immense treasure. Dracula himself was buried at an equal distance from each wife's grave. Presumably, this was to ensure that there would not be any jealousy among the wives in the afterlife.

Ignacio and Zain, vampires themselves, are also experienced treasure hunters, now on a mission to find Dracula's treasure. Ignacio has mapped out the locations of the wives' graves.

Coordinate plane with the points B1, B2, and B3 marked
At what coordinates on the map should vampire-turned treasure hunters Ignacio and Zain begin their treasure hunt? Round to the nearest integer(s).

Our intel tells us that the wives' graves are equidistant to Dracula's grave. If we connect the three locations, B1, B2, and B3, we get a triangle. The circumcenter of this triangle is where the count has been buried.

To find the circumcenter, we can draw the perpendicular bisector of at least two of the triangle's sides.

Drawing a Perpendicular Bisector

There are two steps to construct a perpendicular bisector. Let's remove the coordinate plane for now, leaving only the triangle.


Finding the Circumcenter

If we draw the perpendicular bisector of at least two of the triangle's sides, we can locate the circumcenter which is where these lines intersect.

As we can see, vampires Ignacio and Zain should go to the coordinates (7,7) on the map.

Point is inside and is also equidistant from and On which of the following segments is ? Explain your reasoning.

The exercise tells us that Q is located inside △ ABC. Therefore, we can immediately discard A, C, and E as these options describe the sides of the triangle. Next, let's make a diagram of an arbitrary triangle including the perpendicular bisectors of each side.

We know that Q is on one or more of these perpendicular bisectors. But which one(s)? To answer that, we recall the Perpendicular Bisector Theorem.

Perpendicular Bisector Theorem |- Any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the line segment.

In this case we know that Q is equidistant from A and B. Therefore, according to the theorem, Q must be located along the perpendicular bisector of AB.

Notice that if Q coincides with the circumcenter, then it is actually on all three perpendicular bisectors. But the only perpendicular bisector Q must be on is that of AB. Therefore, the correct option is B.

Inscribed and Circumscribed Circles of a Triangle

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