Lesson Explainer: Constructing Circles Mathematics

In this explainer, we will learn how to construct circles given one, two, or three points.

Recall that, mathematically, we define a circle as a set of points in a plane that are a constant distance from a point in the center, which we usually denote by 𝑀.

A line segment from the center of a circle to the edge is called a radius of the circle, which we have labeled here to have length π‘Ÿ.

To begin with, let us consider the case where we have a point 𝐴 and want to draw a circle that passes through it. Keep in mind that to do any of the following on paper, we will need a compass and a pencil.

How To: Constructing a Circle given One Point on It

  1. To begin, let us choose a distinct point 𝑀 to be the center of our circle. This point can be anywhere we want in relation to 𝐴.
  2. Next, we need to take a compass and put the needle point on 𝑀 and adjust the compass so the other point (holding the pencil) is at 𝐴. The distance between these two points will be the radius of the circle, π‘Ÿ.
  3. Finally, we move the compass in a circle around 𝑀, giving us a circle of radius π‘Ÿ.

As we can see, the process for drawing a circle that passes through 𝐴 is very straightforward. Since we can pick any distinct point 𝑀 to be the center of our circle, this means there exist infinitely many circles that go through 𝐴.

A natural question that arises is, what if we only consider circles that have the same radius (i.e., congruent circles)? That is, suppose we want to only consider circles passing through 𝐴 that have radius π‘Ÿ. If the radius of a circle passing through 𝐴 is equal to π‘Ÿ, that is the same as saying the distance from the center of the circle to 𝐴 is π‘Ÿ. Thus, if we consider all the possible points where we could put the center of such a circle, this collection of points itself forms a circle around 𝐴 as shown below.

Any circle we draw that has its center somewhere on this circle (the blue circle) must go through 𝐴. We demonstrate this with two points, π‘€οŠ¨ and π‘€οŠ©, as shown below.

As we can see, all three circles are congruent (the same size and shape), and all have their centers on the circle of radius π‘Ÿ that is centered on 𝐴. As a matter of fact, there are an infinite number of circles that can be drawn passing through a single point, since, as we can see above, the centers of those circles can be placed anywhere on the circumference of the circle centered on that point.

Now, what if we have two distinct points, and want to construct a circle passing through both of them? We note that any circle passing through two points has to have its center equidistant (i.e., the same distance) from both points. We can use this fact to determine the possible centers of this circle. Hence, we have the following method to construct a circle passing through two distinct points.

How To: Constructing a Circle given Two Distinct Points on It

Let us start with two distinct points 𝐴 and 𝐡 that we want to connect with a circle.

These points do not have to be placed horizontally, but we can always turn the page so they are horizontal if we wish.

  1. First, we draw the line segment from 𝐴 to 𝐡.
  2. Next, we find the midpoint 𝑀 of this line segment.
  3. Now, let us draw a perpendicular line, going through 𝑀.
  4. Choose a point on the line, say π‘€οŠ¨. We then construct a circle by putting the needle point of the compass at π‘€οŠ¨ and the other point (with the pencil) at either 𝐴 or 𝐡 and drawing a circle around π‘€οŠ¨. This is shown below.

We note that any point on the line perpendicular to 𝐴𝐡 is equidistant from 𝐴 and 𝐡. So if we take any point on this line, it can form the center of a circle going through 𝐴 and 𝐡.

We note that since we can choose any point on the line to be the center of the circle, there are infinitely many possible circles that pass through two specific points. We demonstrate some other possibilities below.

As we can see, the size of the circle depends on the distance of the midpoint away from the line 𝐴𝐡. Let us see an example that tests our understanding of this circle construction.

Example 1: Recognizing Properties of Circle Construction

Consider the two points 𝐴 and 𝐡. What is the radius of the smallest circle that can be drawn in order to pass through the two points?

Answer

Recall that every point on a circle is equidistant from its center. Therefore, the center of a circle passing through 𝐴 and 𝐡 must be equidistant from both. We also recall that all points equidistant from 𝐴 and 𝐡 lie on the perpendicular line bisecting 𝐴𝐡. Hence, the center must lie on this line. Taking 𝑀 to be the bisection point, we show this below.

The radius of any such circle on that line is the distance between the center of the circle and 𝐴 (or 𝐡). We demonstrate this below.

Here, we see four possible centers for circles passing through 𝐴 and 𝐡, labeled 𝑀, π‘€οŠ¨, π‘€οŠ©, and 𝑀οŠͺ. Their radii are given by π‘Ÿ, π‘ŸοŠ¨, π‘ŸοŠ©, and π‘ŸοŠͺ. We note that the points that are further from the bisection point 𝑀 (i.e., π‘€οŠ© and 𝑀οŠͺ) have longer radii, and the closer point π‘€οŠ¨ has a smaller radius. We can see that the point where the distance is at its minimum is at the bisection point 𝑀 itself. If we drew a circle around this point, we would have the following:

Here, we can see that radius π‘Ÿ is equal to half the distance of 𝐴𝐡. So, using the notation that 𝐴𝐡 is the length of 𝐴𝐡, we have π‘Ÿ=12𝐴𝐡.

We have now seen how to construct circles passing through one or two points. We can then ask the question, is it also possible to do this for three points?

Recall that for the case of circles going through two distinct points, 𝐴 and 𝐡, the centers of those circles have to be equidistant from the points. For three distinct points, 𝐴, 𝐡, and 𝐢, the center has to be equidistant from all three points. Thus, in order to construct a circle passing through three points, we must first follow the method for finding the points that are equidistant from two points, and do it twice. Specifically, we find the lines that are equidistant from two sets of points, 𝐴 and 𝐡, and 𝐡 and 𝐢 (or 𝐴 and 𝐢). We then find the intersection point of these two lines, which is a single point that is equidistant from all three points at once.

Let us demonstrate how to find such a center in the following β€œHow To” guide.

How To: Constructing a Circle given Three Points

  1. Let us begin by considering three points 𝐴, 𝐡, and 𝐢.
  2. Draw line segments between any two pairs of points. Here we will draw line segments from 𝐴 to 𝐡 and from 𝐴 to 𝐢 (but we note that 𝐡 to 𝐢 would also work).
  3. Find the midpoints of these lines. We will designate them by π‘€οŠ§ and π‘€οŠ¨.
  4. As before, draw perpendicular lines to these lines, going through π‘€οŠ§ and π‘€οŠ¨. If possible, find the intersection point of these lines, which we label π‘€οŠ©.
  5. Finally, put the needle point at π‘€οŠ©, the center of the circle, and the other point (with the pencil) at 𝐴, 𝐡, or 𝐢, and draw the circle.

We note that since two lines can only ever intersect at one point, this means there can be at most one circle through three points. This fact leads to the following question.

Example 2: Recognizing Facts about Circle Construction

True or False: If a circle passes through three points, then the three points should belong to the same straight line.

Answer

First of all, if three points do not belong to the same straight line, can a circle pass through them? Let us consider the circle below and take three arbitrary points on it, 𝐴, 𝐡, and 𝐢.

Here, we can see that although we could draw a line through any pair of them, they do not all belong to the same straight line. So immediately we can say that the statement in the question is false; three points do not need to be on the same straight line for a circle to pass through them.

What would happen if they were all in a straight line? Let us take three points on the same line as follows.

If we apply the method of constructing a circle from three points, we draw lines between them and find their midpoints to get the following.

Next, we draw perpendicular lines going through the midpoints π‘€οŠ§ and π‘€οŠ¨.

Here, we can see that the points equidistant from 𝐴 and 𝐡 lie on the line bisecting 𝐴𝐡 (the blue dashed line) and the points equidistant from 𝐡 and 𝐢 lie on the line bisecting 𝐡𝐢 (the green dashed line). However, this leaves us with a problem. Since the lines bisecting 𝐴𝐡 and 𝐡𝐢 are parallel, they will never intersect. Hence, there is no point that is equidistant from all three points. This shows us that we actually cannot draw a circle between them.

In conclusion, the answer is false, since it is the opposite. If a circle passes through three points, then they cannot lie on the same straight line.

This example leads to the following result, which we may need for future examples.

Rule: Constructing a Circle through Three Distinct Points

We can construct exactly one circle through any three distinct points, as long as those points are not on the same straight line (i.e., the points must be noncollinear).

Let us further test our knowledge of circle construction and how it works.

Example 3: Recognizing Facts about Circle Construction

True or False: A circle can be drawn through the vertices of any triangle.

Answer

Recall that we can construct one circle through any three distinct points provided they do not lie on the same straight line. It is also possible to draw line segments through three distinct points to form a triangle as follows.

This is possible for any three distinct points, provided they do not lie on a straight line. If they were on a straight line, drawing lines between them would only result in a line being drawn, not a triangle.

Now recall that for any three distinct points, as long as they do not lie on the same straight line, we can draw a circle between them. Thus, we have the following:

  • A triangle can be deconstructed into three distinct points (its vertices) not lying on the same line.
  • We can draw a circle between three distinct points not lying on the same line.

Thus, we can conclude that the statement β€œa circle can be drawn through the vertices of any triangle” must be true.

This example leads to another useful rule to keep in mind.

Rule: Drawing a Circle through the Vertices of a Triangle

For every triangle, there exists exactly one circle that passes through all of the vertices of the triangle. This is known as a circumcircle.

With the previous rule in mind, let us consider another related example.

Example 4: Understanding How to Construct a Circle through Three Points

In the following figures, two types of constructions have been made on the same triangle, 𝐴𝐡𝐢. Which point will be the center of the circle that passes through the triangle’s vertices?

Answer

Recall that for every triangle, we can draw a circle that passes through the vertices of that triangle. For the construction of such a circle, we can say the following:

  1. The center of that circle must be equidistant from the vertices, 𝐴, 𝐡, and 𝐢.
  2. We can find the points that are equidistant from two pairs of points by taking their perpendicular bisectors.
  3. Taking the intersection of these bisectors gives us a point that is equidistant from 𝐴, 𝐡, and 𝐢.

We see that with the triangle on the right: the sides of the triangle are bisected (represented by the one, two, or three marks), perpendicular lines are found (shown by the right angles), and the circle’s center 𝐹 is found by intersection. Since this corresponds with the above reasoning, 𝐹 must be the center of the circle.

For the triangle on the left, the angles of the triangle have been bisected and point 𝑁 has been found using the intersection of those bisections. However, this point does not correspond to the center of a circle because it is not necessarily equidistant from all three vertices.

Thus, the point that is the center of a circle passing through all vertices is 𝐹.

For our final example, let us consider another general rule that applies to all circles.

Example 5: Determining Whether Circles Can Intersect at More Than Two Points

True or False: Two distinct circles can intersect at more than two points.

Answer

It is assumed in this question that the two circles are distinct; if it was the same circle twice, it would intersect itself at all points along the circle.

Let us consider all of the cases where we can have intersecting circles. For starters, we can have cases of the circles not intersecting at all.

The circles could also intersect at only one point, 𝑃.

Also, the circles could intersect at two points, π‘ƒοŠ§ and π‘ƒοŠ¨.

Is it possible for two distinct circles to intersect more than twice? Let us suppose two circles intersected three times. That means there exist three intersection points π‘ƒοŠ§, π‘ƒοŠ¨, and π‘ƒοŠ©, where both circles pass through all three points. Recall that we know that there is exactly one circle that passes through three points π‘ƒοŠ§, π‘ƒοŠ¨, and π‘ƒοŠ© that are not all on the same line. Since there is only one circle where this can happen, the answer must be false, two distinct circles cannot intersect at more than two points.

Let us finish by recapping some of the important points we learned in the explainer.

Key Points

  • We can draw any number of circles passing through a single point 𝐴 by picking another point 𝑀 and drawing a circle with radius equal to the distance between the points.
  • We can draw any number of circles passing through two distinct points 𝐴 and 𝐡 by finding the perpendicular bisector of the line 𝐴𝐡 and drawing a circle with center π‘€οŠ¨ that lies on that line.
  • The smallest circle that can be drawn through two distinct points 𝐴 and 𝐡 has its center on the line segment from 𝐴 to 𝐡 and has radius equal to 12𝐴𝐡.
  • We can draw a single circle passing through three distinct points 𝐴, 𝐡, and 𝐢 provided the points are not on the same straight line. We do this by finding the perpendicular bisector of 𝐴𝐡 and 𝐴𝐢, finding their intersection, and drawing a circle around that point passing through 𝐴, 𝐡, and 𝐢.
  • Two distinct circles can intersect at two points at most.

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