Assignment 8

Altitudes and Orthocenters

by Jeff Hall

Problem 1

Construct any triangle ABC.

Problem 2

Construct the Orthocenter H of Triangle ABC

Problem 3

Construct the Orthocenter of Triangle HBC.

The orthocenter of Triangle HBC is Point A, regardless of the shape of Triangle HBC.
Problem 4

Construct the Orthocenter of Triangle HAB.

The ortocenter of Triangle HAB is Point C.

Again, this is without regard to the size or shape of Triangle HAB.

Problem 5

Construct the orthocenter of Triangle HAC.

The orthocenter of Triangle HAC is Point B in all cases.
Problem 6

Construct the circumcircles of Triangles ABC, HBC, HAB, and HAC.

Problem 9

Construct triangle ABC, its incircle, its three excircles, and its nine-point circle. Conjecture? Proof?

Here areTriangle ABC, it's incircle (black), its three incircles (red), and its nine-point circle (blue) :

Any similarities between this drawing and a well-known cartoon rodent are purely intentional.
What can we say about these circles? First, let's draw rays between the center of the nine-point circle

to the center of the excircles. Also, let's draw a raw between the nine-point circle radius and the incircle radius:

Notice that the rays from the nine-point circle cross the point of tangency between the nine-point circle

and the excircles. This occurs regardless of the shape of Triangle ABC, so the excircles and the nine-point circle are tangent.