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

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

Can you prove this for any triangle? These might include some of the following points of concurrency click for a GSP sketch illustration: The Euler Line is the result of connecting the point of pount of the orthocenter, centroid, and circumcenter. A possible gdrgonne to the Euler Line can now be explored fro the Gergonne Point.

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. Notice the relationship of the triangles in the figure and the greater than or gergojne to 4 ratio. 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. Gergonne point These linear equations might be explored on software such as Algebra Xpressor.

A key question that might be raised by students is whether this point of concurrency occurs for any shape of triangle. Assume that there are at least two points of intersection between the lines.

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## Joseph Diez Gergonne

Life[ edit ] In , Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion because the French government feared a foreign invasion intended to undo the French Revolution and restore Louis XVI to the throne of France. He saw action in the major battle of Valmy on 20 September He then returned to civilian life but soon was called up again and took part in the French invasion of Spain in Gergonne was appointed to the chair of astronomy at the University of Montpellier in

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## Cercles inscrit et exinscrits d'un triangle

However, he died when Joseph was twelve years old. In the French Assembly was at a difficult stage trying to stabilise the country following the French Revolution. After the King was returned to Paris, the Assembly reinforced the frontiers of France by calling for , volunteers from the National Guard. Gergonne gave his support becoming a captain in the National Guard. In April France went to war against Austria and Prussia. The French attack was quickly halted and then Prussian forces invaded France.

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## Joseph Diaz Gergonne

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. Assume that there are at least two points of intersection between the lines. These can be shown similar through gergonnee interior angles and vertical angles which lead to AA Similarity see figure below. The following sketch shows the Euler Line with the Gergonne Point. A key question that might be raised by students is whether this point of concurrency occurs for any shape of triangle.

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