Gergonne point. From Wikipedia, the free encyclopedia. Redirect page. Jump to navigation Jump to search. Redirect to: Incircle and excircles of a. The point of intersection of the lines joining the vertices of a triangle to the points where the sides of the triangle opposite to the vertices touch. Gergonne Point Theorem, Proof using Ceva’s Theorem – Antonio Gutierrez. Triangle, inscribed circle, and Concurrent Cevians. Geometry problems involving .
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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. This is a very informal way to illustrate concurrency – go to the gertonne section for more rigor using Ceva’s Theorem.
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.
Gergonne Point of Triangle ABC
Notice the relationship of the ooint in the figure and the greater than or equal to 4 ratio. This time a GSP sketch can be examined with a corresponding table of values which demonstrate that the Hergonne Point H is between each segment that connects the vertices to the incenter’s points of tangencies. 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.
Thus D’ and D have to be the same point which contradicts the assumption that there could be two distinct points of concurrency. A poont question that might be raised by students is whether this point of concurrency occurs for any shape of triangle. Can you prove this for any triangle?
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. 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 gergonnf created by the inscribed circle of the incenter.
These linear equations might be explored on software such as Algebra Xpressor. The Euler Line is the result of connecting the point of concurrency of the orthocenter, centroid, and circumcenter. These might include some of the following points of concurrency click for a GSP sketch illustration: A possible relationship to the Euler Line can now be explored fro the Gergonne Point.
Gergonne point – Wikipedia
Can you find any other special qualities of the Gergonne Point? The following sketch shows the Euler Line with the Gergonne Point. Now that it has been shown that the point B is between the gergonje 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 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.
The point of intersection of the three lines should be gerognne ordered pair solution to the system of equations which would indicate the existence of the Gergonne Point.
Most geometry students are familiar with the several points of concurrency and the steps necessary to construct such points. Assume that there are at least two points of intersection between the lines.