High School Geometry Unlocked (2016)

GOALS

By the end of this chapter, you will be able to:

•Construct the incenter, incircle, circumcenter, and circumcircle for triangles and polygons

•Construct the centroid and orthocenter for triangles

•Understand the standard equation of a circle and use the equation to solve for the center, radius, or coordinates on a circle

•Use the “completing the square” technique for a circle equation in expanded form

•Graph a circle from its equation

•Apply your understanding to general theorems and proofs

Lesson 8.1. Constructions

In this lesson, you will learn how to construct inscribed and circumscribed circles on triangles and quadrilaterals.

Supplies

You should have your
compass and straightedge
ready for this lesson.

CIRCUMCENTER/CIRCUMCIRCLE

All triangles can have circumscribed circles (also known as circumcircles). The circumcenter of a triangle is the intersection of the perpendicular bisectors of its sides.

This lesson will use
several of the basic
constructions that you
learned in Lesson 2.3
of this book. You
may find it helpful
to refer to that
lesson as needed.

To find the circumcenter, you’ll construct the perpendicular bisectors and see where they meet.

When three or more lines intersect, the intersection point is known as a point of concurrency.

Then, to construct the circumcircle, you’ll center the circle at the constructed circumcenter, and make the circle touch the triangle’s vertices.

Here’s how to do it.

Construct the circumscribed circle of triangle ABC.

Supplies

Access your student
tools to download
larger, printable
versions of the images
in this section.

To find the circumcenter of the triangle, construct the triangle’s perpendicular bisectors. It will be sufficient to construct just two of the three bisectors, since that will be enough to show the point of concurrency.

1. Construct the perpendicular bisector for side AB.

With the compass needle on vertex A and the drawing point on B, make a circle.

Then, with the compass needle on B and the drawing point on A, make another circle, which will intersect the first circle in two places.

Next, use a straightedge to connect the two points where the arcs intersect, as shown. This new line is the perpendicular bisector for AB.

2. Construct the perpendicular bisector for side BC.

Repeat the same process to construct the perpendicular bisector for side BC. You may find it helpful to erase some of the previous arcs so that you can better see what you’re doing.

We have found the circumcenter of triangle ABC. For this exercise, we’ll label the circumcenter as O.

3. Draw the circumcircle.

Next, make a circle that touches the vertices of the triangle. Place the compass needle on point O and the drawing point on one of the vertices. Make a circle. If you did it correctly, you’ll find that the circle touches each of the triangle’s vertices.

This method works to construct a circumcircle for any triangle. Try it on the ones below! Then, try a few more on your own for practice.

Special Case: Right Triangle

In a right triangle, the circumcenter will always be the midpoint of the hypotenuse. You can construct the circumcenter more easily by bisecting the hypotenuse.

INCENTER/INCIRCLE

All triangles can have inscribed circles (also known as incircles). The incenter of a triangle is the point where the angle bisectors intersect. To find the incenter, you’ll construct the angle bisectors of the triangle and see where they meet.

The incircle will be tangent to the triangle’s sides. To find one of these points of tangency, you’ll construct a perpendicular line that passes through the incenter.

Here’s how to do it.

Construct the inscribed circle of triangle ABC.

To find the incenter of the triangle, construct the triangle’s angle bisectors. We only need to construct two of the three angle bisectors, since that will be enough to show the point of concurrency.

1. Construct the angle bisector for angle A.

With the compass needle on vertex A, make an arc that passes through the two legs of that angle. For the purposes of this exercise, we’ll call these two intersection points M and N.

Now, position the compass needle point on M and the drawing point on A. With this radius, make a circle.

Repeat for point N, making a circle of the same radius, which will intersect the previous circle.

Finally, use a straightedge to draw a line through this new intersection and point A.

2. Construct the angle bisector for angle B.

Repeat the same process to construct the angle bisector for angle B. You may find it helpful to erase some of the previous arcs so that you can better see what you’re doing.

We have found the incenter of triangle ABC. For this exercise, we’ll label the incenter as O.

Next, make a circle that is tangent to the sides of the triangle. In order to make sure that it’s tangent, construct a line perpendicular to one of the sides. You only need to do this once; this will correctly identify the radius of our circle.

3. Construct a perpendicular line through the incenter.

With the compass needle on the incenter O, make an arc that intersects side AB in two places. For this exercise, we’ll call these intersection points P and Q.

Now, with the compass needle point on P and the drawing point on O, make a circle.

Repeat for point Q, making a circle of the same radius, which will intersect the previous circle.

Use the straightedge to draw a line through this new intersection and point O.

4. Draw the incircle.

This perpendicular intersection between the new line and side AB is one of the three tangent points for our circle. Now, the circle’s center and radius are defined, and the circle can be drawn.

Position the compass needle at point O, and the drawing point at the perpendicular intersection that you have just drawn. Finally, make a circle. If you did it correctly, you’ll find that the circle is tangent to each of the three sides of the triangle.

This method works to construct an incircle for any triangle. Try it on the ones below! Then, try a few more on your own for practice.

THE CENTROID AND ORTHOCENTER

Other than the circumcenter and incenter, mathematicians throughout history have identified many, many different definitions of a circle’s “center.” Many of these definitions are obscure and rarely used, but there are two that are quite common: the centroid and orthocenter.

Centroid

The centroid is the intersection of the medians of a triangle. A median is a segment from the triangle’s vertex to the midpoint of the opposite side.

A fun fact about the centroid is that it is known as the triangle’s “center of mass.” In other words, if you had a physical triangle made out of wood or some other material, you could actually balance the triangle on a pencil point that is placed at the centroid. Likewise, if you hung such a triangle from a string that is placed at the centroid, it would balance there as well.

To construct a median of a triangle, first bisect one of the triangle’s sides. (See “Perpendicular Bisector” in Lesson 2.3.) Then, connect the midpoint to the opposite vertex. Construct the remaining two medians to find the centroid.

Try it on the triangles below!

Orthocenter

The orthocenter is the intersection of the altitudes of a triangle. An altitude is a segment from the triangle’s vertex that is perpendicular to the opposite side. The orthocenter can be outside the triangle.

To construct an altitude of a triangle, construct a perpendicular line that passes through the opposite vertex. (See “Perpendicular Line Through a Point” in Lesson 2.3.) Construct two altitudes to find the orthocenter.

Try it on the triangles below!

Special Case: Equilateral Triangle

In an equilateral triangle, the incenter, circumcenter, centroid, and orthocenter are all found at the same point!

CIRCUMCENTER OF A QUADRILATERAL

As mentioned in Chapter 5, not all quadrilaterals have circumcircles. In fact, one of Euclid’s discoveries was that a quadrilateral will have a circumcircle only if its opposite angles are supplementary. You won’t always know the angles of a quadrilateral, so in order to construct a [possible] circumcircle, you may just go through a little trial and error.

If a quadrilateral (or other polygon) does have a circumcenter, then its definition is the same as that for a triangle—the circumcenter is the intersection of the perpendicular bisectors of the polygon’s sides. To attempt construction of a polygon’s circumcircle, construct each of the perpendicular bisectors of its sides. If and only if the perpendicular bisectors intersect at a single point, then the circumcircle can be constructed.

Construct the circumcircle for quadrilateral ABCD.

First, we can tell you that this quadrilateral does in fact have a circumcenter. To find it, construct the perpendicular bisectors of the quadrilateral’s sides, and see where they intersect. For this exercise, we will assume that you are comfortable with constructing perpendicular bisectors. Again, refer to Lesson 2.3 if needed!

1. Construct the perpendicular bisectors for each side.

Construct the perpendicular bisector for AB:

If you know that a
quadrilateral has a
circumcenter, then you
only need to construct
two perpendicular
bisectors in order to
find the circumcenter.
It’s more common that
you wouldn’t know
whether or not the
circumcenter can be
constructed, so make
a habit of constructing
all four perpendicular
bisectors.

Construct the perpendicular bisector for BC:

Construct the perpendicular bisector for CD:

Construct the perpendicular bisector for DA:

We have found the circumcenter of the quadrilateral.

2. Draw the circumcircle.

Finally, construct the circle centered at the circumcenter and touching the vertices of the quadrilateral.

INCENTER OF A QUADRILATERAL

As mentioned in Chapter 5, not all quadrilaterals have incircles. One property of quadrilaterals that have incircles is that the two pairs of opposite sides have the same total length. You won’t always know the side lengths of a quadrilateral, so in order to construct a [possible] incircle, you may just go through a little trial and error.

If a quadrilateral (or other polygon) does have an incircle, then its definition is the same as that for a triangle—the incenter is the intersection of the angle bisectors of the polygon. To attempt construction of a polygon’s incircle, construct each of the angle bisectors. If and only if the angle bisectors intersect at a single point, then the incircle can be constructed.

Construct the incircle for quadrilateral ABCD.

First, we can tell you that this quadrilateral does in fact have an incenter. To find it, construct the angle bisectors of the quadrilateral, and see where they intersect. For this exercise, we will assume that you are comfortable with constructing angle bisectors. Again, refer to Lesson 2.3 if needed!

1. Construct the angle bisectors for each angle.

Construct the angle bisector for A:

If you know that a
quadrilateral has an
incenter, then you only
need to construct two
angle bisectors in order
to find the incenter.
It’s more common
that you wouldn’t
know whether or not
the incenter can be
constructed, so make
a habit of constructing
all four angle bisectors.

Construct the angle bisector for B:

Construct the angle bisector for C:

Construct the angle bisector for D:

We have found the incenter of quadrilateral ABCD. For this exercise, we’ll label the incenter as O.

Next, make a circle that is tangent to the sides of the quadrilateral. In order to make sure that it’s tangent, construct a perpendicular line to one of the sides. You need to do this only once; this will correctly identify the radius of the circle.

2. Construct a perpendicular line through the incenter.

With the compass needle on the incenter O, make an arc that intersects side AB in two places. For this exercise, we’ll call these intersection points P and Q.

Now, position the compass needle point on P and the drawing point on O. With this radius, sweep the compass around and make an arc that’s outside of the quadrilateral, on the opposite side of AB.

Repeat for point Q, making an arc of the same radius, which will intersect the previous arc.

Use the straightedge to draw a line through this new intersection and point O.

3. Draw the incircle.

This perpendicular intersection between the new line and side AB is one of the four tangent points for our circle. Now, the circle’s center and radius are defined and the circle can be drawn.

Position the compass needle at point O, and the drawing point at the perpendicular intersection that you have just drawn. Finally, make a circle. If you did it correctly, you’ll find that the circle is tangent to each of the four sides of the quadrilateral.