Do we live in an ORTHOCENTRIC System?!
By: Carly Cantrell
LetŐs begin this exploration by making the following
constructions:
1. Construct 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.
The Orthocenter for triangle ABC is H.
The Orthocenter for triangle HBC is A.
The Orthocenter for triangle HAB is C.
The Orthocenter for triangle HAC is B.
Wow, see for yourself!
This is no coincidence. In fact, we call this an Orthocentric System.
An orthocentric system is a set of four points, in
this case A, B, C, and H on the same plane. One of which is always the
orthocenter of the triangle formed by the other three points.
Moreover, the four triangles we have created (ABC,
HBC, HAB, and HAC) all share the same nine-point circle.
Below is the construction for the nine-point circle
for triangle ABC:
Below is the construction of the nine-point circle for
triangles ABC, HBC, HAB, and HAC:
The nine-point circle, in red, is the same for the
points in an orthocentric system.
Why is this?
These four triangles have circumcircles with
equivalent radii. I will let U represent the common nine-point center and P is
a random point in the orthocentric system.
Observe, UA2 + UB2 + UC2
+ UD2 = 3R2 (where R is the common circumradius)
Observe, PA2 + PB2 + PC2
+ PD2 = N2 (where N is a constant)
Assume, P has a locus centered at U. Then as P approaches U, the locus for P collapses onto U. Hence, the locus of P is the nine-point circle.
Observe, PA2 + PB2 + PC2
+PD2 = 4R2