### GERGONNE POINT PDF

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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. 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. Most geometry students are familiar with the several points of concurrency and the steps necessary to construct such points.

These might include polnt of the following points of concurrency click gergpnne a GSP sketch illustration: A key question that might be raised by students is whether this point of concurrency occurs for any shape of triangle.

### Nobbs’ Points, Gergonne Line

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. This time a GSP sketch can be examined with a corresponding table of values which demonstrate that the Gergonne Point H is between each gwrgonne that connects the vertices to the incenter’s points of tangencies.

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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. This is a very informal way to illustrate concurrency – go to the next section for more rigor using Ceva’s Theorem.

Ceva’s Theorem proves concurrency for a point by examining similar triangles and certain ratios that equal 1 see picture below. These can be shown similar through alternate interior angles and vertical angles which lead to AA Similarity see figure below.

Assume that there are at least two points of intersection between the lines. Thus D’ and D have to be the same point which contradicts the assumption that there could be two distinct points of concurrency. 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 poimt.

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These linear equations might be explored on software such as Algebra Xpressor. 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.

## Gergonne point

gergojne A possible relationship to the Euler Line can now be explored fro the Gergonne Point. The Euler Line is the result of connecting the point of concurrency of the orthocenter, centroid, and circumcenter.

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

## Gergonne Point

Notice the relationship of the triangles in the figure and the greater than or equal to 4 ratio. Can you prove this for any triangle?

Can you find any other special qualities of the Gergonne Point?