Dynamic Geometry Module
Lesson 5: The Center of Things

Centers We Know and Love

The medians of a triangle all intersect in a point (the centroid of the triangle). The same is true of the angle bisectors (the incenter), the altitudes (the orthocenter) and the perpendicular bisectors of the sides (the circumcenter). These are all examples of important "centers" for a triangle.

Use Sketchpad to construct each of these centers.

Napoleon's Configuration

This is also referred to as the Toricelli Configuration. It consists of an equilateral triangle drawn outward from each side of a triangle. You can see an example in the accompanying sketch for this lesson (See file ex5_1.gsp).

In this sketch, P, Q, and R are the vertices of the equilateral triangles and X, Y, and Z are their centroids (which also happen to be the circumcenters, incenters and orthocenters!). Napoleon's Theorem states that the centers X, Y, and Z form an equilateral triangle. Join these points and measure the sides of the resulting triangle to verify this. Be sure to move the vertices around and see that this property holds in all cases.

Now delete those segments and join each center to the vertex of the original triangle that is opposite it. That is, draw the lines AX, BY, and CZ. Notice that these lines meet in a point. This point is called the First Napoleon Center for the triangle. Is it always inside the original triangle? If not, can you determine under what conditions it will be outside?

Delete these lines and construct the lines AP, BQ, and CR. These also meet in a point. This is called the First Brocard Point for the triangle. Is it always inside the triangle? If not, can you determine when it is outside? Construct this intersection point and label it O. Measure the angles AOB, BOC, and COA. What do you find? Does this hold when you change the original triangle?

The Gergonne Point

Open a new sketch and draw an arbitrary triangle ABC. Draw the inscribed circle and construct the point, X, where this circle touches BC; Y where this circle touches AC; Z where this circle touches AB. Then draw the lines AX, BY, and CZ. They meet in the Gergonne Point.

Save this sketch. You will need it for your homework.

The Nagel Point

Let X be the point where the ex-circle opposite A touches BC; Y the point where the ex-circle opposite B touches AC; Z the point where the ex-circle opposite C touches AB.

The Nagel Point is the intersection of the lines AX, BY, and CZ. Beneath the figure for the Gergonne Point, create a figure for the Nagel point. Save the resulting sketch.

And There's More!

There are almost as many such centers as you can imagine. The homework exercises ask you to draw diagrams for a few more and to invent some of your own!

For instance, here are some interesting centers:

  1. The intersections of the symmedians. A symmedian is the line you get by reflecting a median across the angle bisector at its vertex. This is the Lemoine Point.

  2. The angle bisectors of the triangle formed by the midpoints of the sides of a triangle meet at the Spieker Center of the triangle.

  3. The orthic triangle is the triangle formed by the feet of the altitudes. The tangential triangle is the triangle formed by the tangents to the circumcircle at the vertices. These two new triangles have parallel sides! When corresponding vertices are joined, the resulting lines meet is a point. This point, sad to say, has no particular name.

Homework

Complete the exercises in Lesson 5 homework and submit them to the MTL handin system.