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MathKit 6

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Remarkable Points and Lines in a Triangle

The triangle is both the simplest and the most thoroughly studied geometric figure. It has an abundance of interesting points, lines, circles, and other curves intricately related to one another. Here we present some of these remarkable objects.

Euler Line and Nine-Point Circle

Many objects and facts connected with the triangle were discovered by Leonhard Euler, one of the greatest mathematicians of all time, and were named after him. Euler proved that three points in a triangle, the center O of its circumcircle, called the circumcenter, the meet G of its medians, called the centroid, and the meet H of its altitudes, called the orthocenter, lie on the same line and that in any non-equilateral triangle point G divides the segment OH in the same ratio OG:GH=1:2. The line OH was called the Euler line. Later it was found that this line goes through many other remarkable points. For example, the midpoint of OH is the center of the circle that passes through the midpoints of the triangle's sides, through the feet of its altitudes, and through the midpoints of the segments joining the orthocenter to the vertices. Because of these nine points, this circle is called the nine-point circle. It has other names, too: Feuerbach's circle, Euler's circle, Terquem's circle and so on.

Here you'll find a triangle's dynamic model which shows all the points and lines mentioned above and two activities in which you will construct the Euler line and the nine-point circle yourself.


Learn Euler line and nine-point circle

Construct the Euler line of the given triangle

Construct the nine-point circle of the given triangle

Gergonne Point and Nagel Point

Here you'll get to know two more triangle's points of interest, each of them being the intersection point of three lines. The Gergonne point Ge lies on the lines joining the vertices to the points at which their opposite sides touch the incircle and the Nagel point N is the common point of the lines joining the vertices to the contact points of the opposite sides and the corresponding excircles.


Learn Gergonne point and Nagel point

Construct the Gergonne point of the given triangle

Construct the Nagel point of the given triangle

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