Geometry

Triangle Centers: Circumcenter

Moving forward from perpendicular bisectors, let’s talk about a topic you’ll certainly see in a geometry course: triangle centers.

If you Google “triangle centers“, you’ll find that there are actually compiled indexes of these creatures – they number in the hundreds at least. In fact, there are so many that they have to be named using alphanumeric codes! To keep this site from turning into another exhaustive list, we will cover the four triangle centers that are by far the most common ones you will see.

Part 1: Circumcenter

Hint: We’ll need our knowledge from the last post on perpendicular bisectors to figure out this triangle center.

The circumcenter is the point on the triangle that is equidistant from each of the triangle’s vertices. To put it another way, the lines connecting the circumcenter and each vertex are the same length. So we can say that the circumcenter is the center of a circle (aptly named the circumcircle) that passes through the triangle’s vertices. In the figure below, D is the circumcenter and the center of the circumcircle.

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We’re given a triangle, like this:

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How does one construct a circumcenter? Remember, we’re looking for a point that is equidistant from the triangle vertices. As it happens, any point equidistant from the vertices of a line is on the perpendicular bisector of that line. (Recall the arcs drawn in the last post.) So let’s find the perpendicular bisectors of two of the triangle sides (three, if you want, but you only need two lines to find an intersection).

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In this diagram, we have constructed the perpendicular bisectors of AB and AC. Any point on the bisector of AB is equidistant from A and B, and any point on AC’s bisector is equidistant from A and C. Logically, if we take the intersection of the two bisectors D, then D is equidistant from A, B, and C. Therefore, D is the circumcenter of triangle ABC.

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To summarize: The circumcenter of a triangle is the intersection of two of the triangle’s perpendicular bisectors because the intersection is equidistant from each of the vertices of the triangle.

NEW: If you have a copy of Isosceles on your iPhone, iPod touch, or iPad, you can download the Isosceles GSK file from this post and explore circumcenters on your own. Tap here on your device to download the file.

If this helped you understand perpendicular bisectors, or if you have any questions, please post a comment below. Next time we’ll discuss another triangle center, the incenter.

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Geometry

Perpendicular Bisectors

Armed with our profound understanding of Triangle Angle Sum, let’s march on and conquer the perpendicular bisector. What is a perpendicular bisector, you ask? Math Open Reference says this:

A line which cuts a line segment into two equal parts at 90°.

The Classical Greek mathematicians preferred a minimalist approach to geometry; they only used a compass and straightedge to make constructions. Even with the modern Cartesian grid, the tradition has persisted through the ages, and every geometry student should know and understand how to construct a bisector without shortcuts. Ready?

We’re given a line, such as this one:

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The idea behind this construction is to first construct two isosceles triangles, one on either side of the line. As you know, the two legs of an isosceles triangle are the same length, and an easy way to construct this is by using the compass. Why? Because the compass draws circles, and all the points on a circle are the same distance from the center. Set your compass to more than half of the line length (so the sides won’t fall short), then make an arc on either side of the line from both points – four arcs total.

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The intersections of the arcs will be the vertices of the isosceles triangles, ACB and ADB. (Generally, you won’t have to draw the triangles – I’m just showing the full explanation of why the process works.)

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Now let’s observe our diagram for a second. Two pieces of information are going to be helpful to us here:

  • The sum of the angles in each of the triangles ACB and ADB is 180°, as we know from above.
  • Triangles ACB and ADB are congruent (by SSS, if you’re wondering), and therefore all the corresponding parts of the triangles are congruent.

Next, connect points C and D. Place point E at the intersection of CD and AB. Your figure should look something like this:
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Examining the diagram again, we can tell that CD has bisected (cut in half) the vertex angles ACB and ADB (for a full explanation of this, see below). So the angles from triangle ACB have been divided between the new triangles, ACE and BCE. Therefore each smaller triangle contains half of the angle measure of the larger triangle – 90°. But every triangle has to have 180°, right? We still have 90° left to account for. So we turn to angles AEC and BEC, which, by Triangle Angle Sum, must measure 90°. The same holds for triangle ADB.

Also, since the smaller triangles are congruent to each other, the segments AE and EB must be congruent.

The definition of “perpendicular bisector” states that a perpendicular bisector must cut the segment into two equal parts, which we showed above, and meet the segment at 90°, which we also showed. So CD is the perpendicular bisector of AB!

To summarize, the perpendicular bisector of a line is constructed by drawing two congruent isosceles triangles around the line, then bisecting their vertex angles.

A Full Explanation of Bisecting the Vertex Angles

Triangles ACD and BCD are congruent by SSS – we know each corresponding side of the triangles is congruent, so the whole triangles are congruent. Therefore we know that angles ACD and BCD are congruent by CPCTC; likewise for angles ADC and BDC. By definition of angle bisector, CD bisects angles ACB and ADB.

That should take care of any questions about why any of this logic works, but if you do have a question, please leave a comment so I can get back to you. If I helped you understand perpendicular bisectors, don’t forget to share this post!

Thanks!