There's actually a (usually) easier way to construct a nine-pointer, without even drawing any triangles.
1. Start with your circle.
2. Mark three points anywhere on it (they can all be in the same 180-degree arc, or even have two or three in the same location). These will be the feet of the altitudes.
3. The three points you just marked divide the circle into three arcs, and any two adjacent arcs can be combined to form another arc. Bisect these six arcs; the points bisecting the three basic arcs will be the other altitude crossings, and the points bisecting the combined arcs will be midpoints.
3a. If two of the arcs are the same size, the bisector of the arc combining them will lie on one of the points from step 2, resulting in an 8-pointer. If all three are the same size, all three combined-arc bisectors will lie on the points from step 2, resulting in the 6-pointer.
3b. If two of the points are in the same location, the arc between them is of size 0, so the bisector of that arc is in that location as well. Two of the combined arcs also become identical to basic arcs, so there are only three bisectors, resulting in a five-pointer (or four-pointer, if the third point lies on the bisector of the combined arc).
3c. If all three points were in the same location, three of the bisectors will be there as well, and the other three will be on the opposite side of the circle (making a 2-pointer).
3d. The bisector of a combined arc will be exactly opposite the circle from the bisector of the opposite basic arc; as such, it may be easier to only bisect the arcs less than or equal to 180 degrees, and then mark the points opposite them.