Assignment 8 –Altitudes and Orthocenters
In assignment 8, we were to discover more about triangles, and their special properties.
An orthocenter of a triangle is the point of concurrency for the three possible altitudes of a triangle.
An altitude of a triangle is also known as the height of the triangle.
The point of concurrency at which the base of a triangle and its height meet is called the foot of the perpendicular.
The point of concurrency at which the perpendicular bisectors meet is called the circumcenter.
The circumcenter is center of the circumcircle whose radius is length of the circumcenter of the orthic-triangle to the foot of the perpendicular of the original triangle.
To create the previous picture, follow these steps:
1. Construct any triangle ABC.
2. Construct the Orthocenter H of triangle ABC.
3. Construct the Orthocenter of triangle HBC.
4. Construct the Orthocenter of triangle HAB.
5. Construct the Orthocenter of triangle HAC.
6. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.
What do you notice? Conjectures?
One can see:
a) The orthocenter of triangle HBC is A.
b) The orthocenter of triangle HAB is C.
c) The orthocenter of triangle HAC is B.
The following is an informal proof of a), b), and c).
d) The circumcircles for triangles HBC, HAB, and HAC are the same.
e) The circumcircle of the orthic-triangles of a given triangle is the nine-point circle of the given triangle.
A nine point circle is a circle whose center is the midpoint between the orthocenter and circumcenter of a triangle. The nine points its circumference passes through are the foot of the perpendiculars of each side of the triangle, the point concurrent with a side and the sides perpendicular bisector, (three of them) and the midpoints of the vertices to the orthocenter of the triangle.
The following picture is a view of the nine point circle of the orthic-triangles HBC, HAC, and HAB, along with the original triangle ABC. One can notice that all four circles are the same circle, the solid orange circle. .
Once, created in GSP, if any vertices of the triangle ABC was moved to where the orthocenter H is located, a right triangle ABC is formed, where now the orthocenter H is located at the vertex of the right triangle.
What happens when A becomes the Orthocenter?
What happens when B becomes the Orthocenter?
What happens when C becomes the Orthocenter?