Altitudes and Orthocenters
Perform
the following constructions
a. Construct any triangle
ABC.
b. Construct the
Orthocenter H of triangle ABC.
c. Construct the
Orthocenter of triangle HBC.
d. Construct the
Orthocenter of triangle HAB.
e. Construct the
Orthocenter of triangle HAC.
f. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.
After performing these
directions I got this construction:
I then constructed the nine point circles for
triangles ABC, HBC, HAC, and HAB, found something very interestingÉ
THEIR NINE POINT CIRCLES ARE THE SAME!
But why is this??
Recall that a nine-point circle passes through nine
points: the three midpoints of the sides, the three feet of the altitudes, and
the three midpoints of the segments from the respective vertices to the
orthocenter. The center of a nine-point circle is the midpoint between the
orthocenter and the circumcenter.
The center of this nine-point triangle also lies on
the Euler Line. This Euler Line passes through the orthocenter, circumcenter, and centroid of the triangle. Since triangles
HBC, HAC, and HAB are all within ABC connecting to its orthocenter, these
triangles all have the same nine-point circles.