For any given triangle, multiple circeles can be created that are related
to the triangle. The relationship between the incircle, three excircles
and nine-point circle for a triangle will be examined. A scalene triangle
will be examined, followed by several specific triangles including the equilateral,
isosceles, and right triangles. Click on any of the four triangles to jump
to the investigation of that type of triangle.
To help with the diagrams in the following investigations, the colors
and line styles will be kept consitent. The original triangle has solid
black sides. The excircles of a triangle will be blue, while the incircle
will be green. The nine-point circle is red. The sides of the triangles
have been extended as dashed lines. An example diagram follows.
Depending upon the configuration of the triangle, the incircle can be tangeant to one or more of the excircles as well. The following diagram shows that the incircle can be tangeant to the nine-point circle and one of the excircles.
Through trial and error, it can be shown that the the nine-point circle and incircle can become the same circle. It appears that the triangle shown is an equilateral triangle. Equilateral triangles will be examined next to see if this holds true for a triangle that is constructed as an equilateral triangle.
To examine the five circles for a triangle using GSP, click here.
To examine the five circles for an equilateral triangle using GSP, click here.
To examine the five circles for an isosceles triangle using GSP, click here.
To examine the five circles for a right triangle using GSP, click here.
To go to a library of GSP scripts, click here.