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.