Everyone here is saying that it's not necessary to figure out the exact triangle in the book, and for purely dueling purposes that's true. But it's still worth questioning whether that diagram is any good from a standpoint of plain curiosity.
I'm a couple of years late, but I think I've figured it out (attached image)
I know Lightning kind of already said this, but the points on a nine-point circle are:
The feet of the altitudes (type I points)
The midpoints of the segments connecting the orthocenter to the corners of the triangles (type II points)
The midpoints of the sides of the triangle (type III points)
These types aren't actual geometric terms, but I'm going to use them to explain my methodology.
It turns out, there's actually a much easier way to construct a nine-point circle without the triangle.
Place three points randomly along a circle. These are the type I points.
Bisect the minor arc between each pair of type I points. These are the type II points.
Find the points diametrically opposed to each type II point. These are the type III points.
With this newfound knowledge, we can jump into our investigation.
Let's consider the attached labeling schema. We can pretty much assume that the point towards which all the grey segments are drawn is the orthocenter of the triangle. The chances that four different pairs of points - AF, BG, DH, and EI - are all colinear with a point chosen inside the circle seems far too unlikely to be purely coincidental. However, if that chosen point is the orthocenter, three pairs of points necessarily have this property, as each type I point must have a corresponding type III point along the same altitude.
Consequently, we can assume that three of AF, BG, DH, and EI must be altitudes, and one is a fake altitude. Meanwhile, C doesn't have an opposing point, so it must not be along an altitude. Therefore, it is a type III point.
Since C is a type III point, the point diametrically opposed to it - G - is a type II point. Thus, the point altitudinally opposed to G - B - must be a type I point.
Also since G is a type II point, we know that it is at the center of a minor arc between two type I points. Arcs HF and IE appear to be the only viable candidates.
Therefore, either HFB or IEB are the sets of type I points. Out of the two, I thought HFB looked more likely.