How many circles can pass through three points?
The mathematical definition of a circle is that it is a set of points in a plane that
are a constant distance from a point in the center. We are going to consider how many circles can pass through three points. So let’s consider any three points, and we can define them as 𝐴, 𝐵, and 𝐶. A circle would pass through these three points if its center is equidistant from the
three points. An example of a circle which wouldn’t go through points would have a center here. A circle with a center here wouldn’t work because we can see that it’s not the same
distance from the center to each of the points 𝐴, 𝐵, and 𝐶.
And so we must ask, how do we find the center of a circle that would pass through
these three points? How do we find the point which is equidistant from these three points? Let’s begin by drawing line segments between any two pairs of points. So here, we’ve got the line segment 𝐴𝐵 and the line segment 𝐵𝐶. The line segment 𝐴𝐶 would also work. We will now construct the perpendicular bisectors of each of these line segments. To do this accurately, we need one of these tools, which will be a compass or a pair
of compasses, depending on where you live.
Let’s start with the perpendicular bisector of the line segment 𝐴𝐵. We take the sharp end of the compass and put it in point 𝐴, and then we stretch it
open so that it’s longer than half of the length of the line segment 𝐴𝐵. Using the pencil, we draw an arc above the line segment and below it. We then repeat the process. This time, we put the pointy end into point 𝐵 and create another set of arcs, one
below the line and one above the line.
We observe that each of the sets of arcs above and below the line segment has a point
of intersection. Drawing the line between these two points of intersection creates the perpendicular
bisector. Notice how this line is at 90 degrees to the line segment 𝐴𝐵, and we have created
two congruent line segments. It’s usually very good to keep in our construction lines, but let’s remove them so
that we can see exactly what’s happening.
The line that we have drawn here represents all the points that are equidistant from
𝐴 and 𝐵. We can now repeat the process by finding the perpendicular bisector of the line
segment 𝐵𝐶. We can draw the first set of arcs from point 𝐵 and the second set from point 𝐶. And we have completed the second perpendicular bisector.
This line represents all the points which are equidistant from point 𝐵 and 𝐶. We don’t need to draw the line segment 𝐴𝐶 and find its perpendicular bisector. And that’s because the point where these two perpendicular bisectors intersect is the
point which is equidistant from 𝐴, 𝐵, and 𝐶. This point can therefore be the center of a circle which passes through the three
points. It would look something like this.
And so how many circles will pass through the three points? Well, we know that there’s one because we’ve drawn it. But will there be any more? We can alternatively ask, are there any other points which are also equidistant from
𝐴, 𝐵, and 𝐶? Remember that the first perpendicular bisector represents all the points equidistant
from 𝐴 and 𝐵. The second bisector represents all the points which are equidistant from 𝐵 and
𝐶. The point where these two lines intersected is the point which is equidistant from
all three points.
So could these two perpendicular bisectors intersect at a different point? How about here, above the three points, or maybe here, below the three points
again? And of course, we should recognize that neither of these is possible. Two straight lines can only intersect at most one point. And therefore, there is only one circle which can pass through three points.
Before we finish by giving the answer, there is one really important thing to
note. This only happens when the three points do not lie on a straight line. If we take the example of 𝐴, 𝐵, and 𝐶 on a straight line, then we can see that
it’s not physically possible to draw a circle through all three points. We could draw a circle that goes through two of the points, but not all three. When the three points lie on a straight line, then zero circles could be drawn
Excluding this case, however, we can give the answer of one circle. Finally, it’s worth mentioning that it’s not just that we might be able to draw a
circle through three points that aren’t on a straight line. But there will always be a circle which we can draw through any three points.