In this assignment I will construct a triangle A,B,C
and it orthocenter H. I will connect the orthocenter to
the vertices with the line segment HA, HB, and HC. I will then
construct the midpoints of HA, HB and HC.
I will connect the midpoints to form a new triangle (DEF)
I will prove that this triangle
(DEF) is similar to the triangle ABC and congruent to it medial
Finally, I will compare the centroid G, the orthocenter H, the circumcenter C, and the incenter I of the triangles.
Below is the construction of the triangle ABC and the triangle
To prove that the triangle ABC is similar to triangle DEF I must show that both triangles
have matching congruent angles ( corresponding angles ).
Note: The distance below are rounded numbers
Also comparing the slope of the lines, we realize that
AB and DE have the same slope as
well as CB and FE, CA and FD
Below is the construction of the medial triangle JKL.
Let me recall you that the MEDIAL triangle is a triangle obtained by connecting the three midpoints of the sides.
It is similar to the original triangle and one-fourth of its area. Construct G, H, C, and I for this new triangle.
From the collected data, you may observe that the side DE is
congruent to side LJ,
the angle E is congruent to angle J,
and the side EF is congruent to side JK.
By SAS theorem, the triangle DEF is congruent to triangle JKL.
Here is another picture to illustrate the congruency of both
You can see that both triangles share the same circumcenter (C) and the rest of the center
points are symmetric to the circumcenter C.
Click here to see the animation of the above construction