An orthic triangle is a triangle that connects the feet of the altitudes
of a triangle. Using Geometer SketchPad(GSP), we will examine the relationships
between the centroid, orthocenter, circumcenter and incenter for a triangle
and its orthic triangle. The first step will be to construct the the orthic
triangle. In the diagram below, the black triangle is the original triangle
and the red triangle is the orthic triangle.
We then need to construct centroid, orthocenter, circumcenter and incenters
for both triangles. The points for the orthic triangle have been labelled
with two letters, the second being O. The centroids are labelled with G,
the orthocenters with H, the circumcenters with C and the incenters with
I. Each construction can be seen below. The colors of the points and their
labels will remain consistent throughout this examination.
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The relationships between these points will be examined for acute triangles
first, then for right and obtuse triangles. When all eight points are constructed
on the same diagram, two of the points overlap. The orthocenter of the original
triangle and incenter of the orthic triangle are the same point for any
acute triangles. An example can be seen below.
When the relationship between the four points was examined for the original triangle, G,H anc C were found to be colinear. This relationship holds for the GO, HO and CO. The diagram below shows this relationship. It also shows that CO is on the line defined by H, G and C. This held true for all acute triangles that were examined.
A special acute triangle that should be examined is the equilateral triangle. As can be seen from the diagram below, all eight points are in the exact same spot for an equilateral triangle.
When the orthic triangle was constructed for a right triangle, it was degenerate. There was only a line from foot of the altitude of the right angle to the right angle. The other two feet were copointer at the vertex of the right angle. Without an orthic triangle, the GO, HO, CO and IO could not be constructed. This result is shown below.
When an orthic triangle is constructed for an obtuse triangle, the legs must be extended to generate all of the altitudes. Once this is done, the orthic triangle can be constructed.
In an obtuse triangle, the orthocenter of the original triangle and incenter of the orthic triangle no longer occupy the same point. After making several obtuse triangles, it became apparent that the location of the incenter was always the vertex of the obtuse angle.
The relation ship between CO and H, G and C was examined. Co continued
to be colinear with H,G, and C.
To examine an orthic triangle using GSP, click
here.
For a scipt that created the orthic triangle and all eight points, click here.
To go to a library of GSP scripts, click
here.