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3. Inscribed and Circumscribed Circles of a Triangle
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eCourses /
Geometry /
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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Lesson Settings & Tools
9 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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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.

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



Level 1
Level 2
Level 3
What are the coordinates of the circumcenter of the triangle with the given vertices?

First we should plot the points and graph the triangle.

The circumcenter is the point of intersection of the perpendicular bisectors of at least two sides of the triangle. Remember that a bisector cuts something in half. Therefore, we must find lines that are perpendicular to the sides at the side's midpoints.

Finding the Circumcenter

A horizontal line is always perpendicular to a vertical line. Because SR is horizontal, any perpendicular line will be vertical. Similarly, we see that RT is vertical. Therefore, any perpendicular line is horizontal. Let's find the midpoint of each side using the Midpoint Formula.

Side M(x_1+x_2/2,y_1+y_2/2) Midpoint
SR U(2+6/2,7+7/2) U(4,7)
RT V(6+6/2,7+1/2) V(6,4)

Let's add these midpoints to our graph. We will also draw the perpendicular bisectors to locate their point of intersection. This is the circumcenter.

The circumcenter is located at (4,4). That means if we place the needle point of a compass at (4,4) then open it until it touches either S, R, or T, we can circumscribe the triangle.

is the circumcenter of Use the given information to find the length of

A triangle ABC with its circumcenter marked at P

From the exercise, we know that P is the circumcenter of △ ABC. By the Circumcenter Theorem, we know that the circumcenter of a triangle is equidistant from its vertices. Therefore, PA, PB, and PC all have the same length.

Since the three segments are congruent, we can equate the expressions we have been given for PA and PC and then solve for x by performing inverse operations until x has been isolated.

When we know the value of x, we can calculate either of the given expressions to find the length of PB. PA: 4( 10)+2 =42 units PC: 5( 10)-8 =42 units This means PB also equals 42 units.

In the figure below, and Find

Triangle ABC with the incenter marked

Examining the diagram, we see that PC bisects ∠ C and PB bisects ∠ B.

Therefore, these segments are angle bisectors and they intersect at the triangle's incenter. By the Incenter Theorem, the incenter of a triangle is equidistant from the sides of the triangle.

Since these three segments are congruent we equate expressions for PD and PE and solve for x.

Now that we know the value of x, we can substitute this into the expressions for PD or PE. If we find their lengths, we also find the length of PF. PD: 5( 3)+8 =23 units PE: 7( 3)+2 =23 units This means PF has a length of 23 units.

Which of the points through on the axis and through on the axis fall on the inscribed circle to It may be helpful to copy the triangle on graphing paper.

Right triangle with one vertex at the origin, one vertex on the positive x-axis, and one vertex on the positive y-axis. 3 points marked on each leg of the triangle.

To inscribe a circle, we need to find the incenter of the triangle. This is the point of intersection of at least two of the triangle's angle bisectors.

Drawing an Angle Bisector

There are four steps needed to construct an angle bisector. Let's remove the coordinate plane for now.


Finding the Incenter

If we draw the angle bisector of at least two of the triangle's sides, we can find the incenter where these lines intersect.

Notice that the incenter is equidistant from every side of the triangle.

Drawing the Inscribed Circle

To inscribe the circle, we also have to find the circle's radius. This segment will be perpendicular to any of the sides of the triangle. To find it, we must again use our compass. There are four steps involved.

When we know where the inscribed circle will intersect one of the triangle's sides, we can open our compass to match this distance and then inscribe the circle.

As we can see, the circle intersects B and E.

Complete the statement with

  • Always
  • Sometimes, or
  • Never.
 The circumcenter of an acute triangle is __________ inside the triangle.
If the perpendicular bisector of triangle's side intersects the opposite vertex, then the triangle is __________ isosceles.

The circumcenter of a right triangle is always on the hypotenuse.

If we increase the right angle, the triangle becomes obtuse. This also moves the circumcenter outside of the triangle.

Conversely, if we decrease the right angle, the triangle becomes acute. The circumcenter then moves inside of the triangle.

Therefore, the missing word is always.

Let's construct a triangle where the perpendicular bisector of one side intersects the opposite vertex of that side.

Let's recall the Perpendicular Bisector Theorem.

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

Since CD is the perpendicular bisector to AB, we know by the theorem that the distance from C to A and the distance from C to B are the same.

As we can see, △ ABC has two congruent sides. Therefore, it is an isosceles triangle and the word we are looking for is always.

Find the coordinates of the centroid of the triangle with the given vertices.

Let's first draw △ FGH in a coordinate plane.

The centroid of a triangle is the point of intersection of the triangle's medians. A median of a triangle is the segment drawn between the midpoint of a triangle's side and its opposite vertex. Therefore, we need to start by calculating the midpoints using the Midpoint Formula.

Side Points M(x_1+x_2/2,y_1+y_2/2) Midpoint
FG ( 4,5), ( 6,9) I(4+ 6/2,5+ 9/2) I(5,7)
GH ( 6,9), ( 8,1) J(6+ 8/2,9+ 1/2) J(7,5)
FH ( 4,5), ( 8,1) K(4+ 8/2,5+ 1/2) K(6,3)

Let's add these midpoints to the diagram.

Finally, we will draw the medians of at least two sides. For good measure, we will draw all three.

The coordinates of the centroid are (6,5).

As in Part A, we will begin by drawing the triangle in a coordinate plane.

In order to draw the triangle's medians we must find the midpoint of each side. For this purpose we will use the Midpoint Formula.

Side Points M(x_1+x_2/2,y_1+y_2/2) Midpoint
AC ( 3,3), ( 6,7) D(3+ 6/2,3+ 7/2) D( 92,5)
BC ( 12,2), ( 6,7) E(12+ 6/2,2+ 7/2) E(9, 92)
AB ( 3,3), ( 12,2) F(3+ 12/2,3+ 2/2) F( 152, 52)

Let's add these midpoints to our graph.

Let's draw the medians of at least two sides

The coordinates of the centroid are (7,4).

Which of the point(s), through are on the circumference of the circumscribed circle to

Triangle XYZ in a coordinate plane with 6 points marked that could possibly be on the circumscribed circle to XYZ

To find the circumcenter, we must draw the perpendicular bisector of at least two of the triangle's sides and then locate their point of intersection.

Drawing a Perpendicular Bisector

To construct a perpendicular bisector there are two steps. For now, we will remove the coordinate plane 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 where these lines intersect.

Circumscribing the Triangle

When we know the center of the circle, we can open our compass to match the distance between the cicrumcenter and one of the triangle's vertices and then draw the circle.

As we can see, B, D, and E are all on the circumference.

Inscribed and Circumscribed Circles of a Triangle

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