The most important picture in triangle geometry

This shows the structures generated by a point in the plane of a triangle. Put a point in the plane and no extra structure occurs. Put two and you may add a line. With a third point you may add two more lines resullting in three points and three lines. When you add a fourth point, P in this figure, look at all that you get simply by adding the lines determined by the points that are present and connecting the points with lines. This structure has a dual nature in that each point corresponds to a line, as shown in the table. A lesson creating and explaining this picture is given below the picture.

The coordinates given are general homogeneous coordinates (which can be learned here ( a pdf file).

Lesson 1 for triangle geometers:

All you need for this lesson is ruler, pencil, paper or a geometry/drawing program. You are about to create the projective structures of a point P in relation to a triangle. Projective structures vary wildly, so if you are using papter and pencil, it might be a good idea to follow the shapes of the triangle above and the placement of P therein.

To create this picture, put 3 points A, B, C in the plane and connet them with edges a, b, c where a the line opposite point A. We have set up both a self dual configuration (the triangle) where sides and points correspond to each other and a notational system to match.

Now put a point P inside the triangle. We now can create three new lines A—P, B—P, C—P. These are called the Cevian lines of P. New lines create new points. The Cevian lines meet the sides of the triangle in points A_P, B_P, C_P. These points are called "traces" of the Cevian lines. We are using John Conway's notation for these points.

Three new points create three new lines a_P, b_P, c_P and a new triangle ∆_P, called the Cevian triangle of P. So each point in the triangle plane (not on the edges of ABC) determines a unique Cevian triangle.

Now it gets interesting. New lines create new points. Each sides of ∆_P meets the corresponding side of ∆. These three points are colinear. This line is called "the tripolar line of P" or the "axis of perspective" of P. In this context P is called "the perspector."

Finally lines from the vertices can be drawn to the corresponding meets from the last step. These are the exCevian lines, which form the ExCevian triangle, sometimes called the preCevian triangle or the antiCevian triangle.

We have produced another self dual situation. Each point and each line in the picture correspond to a line and a point, respectively, as shown in the chart accompanying the picture.

Why is this important? Each point in the plane of the triangle is associated with 7 new lines, the Cevian and exCevian lines and the tripolar line, and two triangles, the Cevian and exCevian one. By itself it is a beautiful structure. In addition we use it all the time.