Circles on the triangle

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Triangle with circumference (red), incircle (green), circles (blue) and Feuerbach circle (purple)

If the geometry of circles on the triangle is mentioned, the following are primarily circles meant that even in the ancient times were studied by Greek mathematicians.

  • The inscribed circle of a triangle is the circle that touches all three sides of the given triangle inside, i.e. the largest circle that lies inside the triangle. The center of the inscribed circle has the same distance on all three sides and consequently lies on all three bisectors (angular symmetries) of the interior angles .
  • The three circles that touch the inside of one side of the triangle and the extensions of the other two sides are defined in a very similar way to the inscribed circle . Their midpoints result from the fact that the bisector of an interior angle is intersected with the bisector of the non-adjacent exterior angle.
  • Another circle with interesting properties that was only discovered in modern times is the Feuerbach circle , named after Karl Wilhelm Feuerbach . It is also referred to as a nine-point circle because it contains the three side centers, the three base points of the triangular heights and the three centers of the "upper height sections" (between the height intersection and the corners). The center of the Feuerbach circle lies on the so-called Euler straight line , which goes through the center of the circumference, the intersection of the heights and the center of gravity . The radius of the Feuerbach circle is half the size of the circumcircle radius. In addition, the Feuerbach circle touches the inscribed circle and the three incoming circles.

See also

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