# GERGONNE POINT PDF

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Most geometry students are familiar with the several points of concurrency and the steps necessary to construct such points. Ceva’s Theorem proves concurrency for a point by examining similar triangles and certain ratios that equal 1 see picture below.

A key question that might be raised by students is whether this point of concurrency occurs for any shape of triangle. The Euler Line is the result of connecting the point of concurrency of the orthocenter, centroid, and circumcenter.

### The Gergonne Point

This is a very informal way to illustrate concurrency – go to the next section for more rigor using Ceva’s Theorem. The point of intersection of the three lines should be the ordered pair solution to the system of equations which would indicate the existence of the Gergonne Point.

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Notice the relationship of the triangles in the figure and the greater than or equal to 4 ratio. The Gergonne Point, so named after the French mathematician Joseph Gergonne, is the point of concurrency which results from connecting the vertices of a triangle to the opposite points of tangency of the triangle’s incircle.

Assume that there are at least two points of intersection between fergonne lines.

Thus D’ and D have to be the same point which contradicts the assumption that there could be two distinct points of concurrency. A possible relationship to the Euler Ppoint can now be explored fro the Gergonne Point.

Can you find any other special qualities of the Gergonne Point? Another approach to showing the existence of the Gergonne Point is to use GSP to create axes and a grid in order to examine the three linear equations that could be formed from making segments that join the vertices of each triangle to the points of concurrency of the incenter to each side.

### Nobbs’ Points, Gergonne Line

These linear equations might be explored on software such as Algebra Xpressor. Although there does not seem to be any special theories concerning the Gergonne point itself, the point can be examined in regard to certain ratios of triangles created by the inscribed circle of the incenter.

Can you prove this for any triangle?

These might include some of the following points of concurrency click for a GSP sketch illustration: Now that it has been shown that the point B is between the other two points on each segment and that the point is on all three segments at the same time, then it must be a point of concurrency for all three segments.

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Click here for a GSP sketch in which a randomly formed triangle can be examined by using the selection tool to move the vertices to change the shape. These can be shown similar through alternate interior angles and vertical angles which lead to AA Similarity see figure below.

## Gergonne point

This essay will prove the existence of this point for any triangle, explore its relationship to the Euler line, if any exist, and discuss the possible usefulness of this point. This time a GSP sketch can be examined with a corresponding table of values which demonstrate that the Gergonne Point H is between each segment that connects the vertices to the incenter’s points of tangencies.

The following sketch shows the Euler Line with the Gergonne Point.