Lets ignore the cross for the moment. Then we simply have a 5 pointed star drawn within a 5 pointed star, drawn within 3 sides of a pentagon.
The interior 5 pointed star is inside a pentagon. There is a known result that says there are 35 triangles in this configuration. There are basically 7 groups with 5 triangles each.
The groups are:
- The points of the star - tall and skinny
- The space between the points - short and fat
- The point paired with the space on its left
- The point paired with the space on its right
- The point and the space on both sides
- Two non-adjacent points including the middle pentagon.
- The space between the points extended to include the point on the opposite side. Makes another tall/skinny triangle that is proportional to #1, but much taller.
If we were to complete the outer pentagon, then there would be another 35 triangles there as well. However, we are missing two of the sides of the pentagon, meaning we are missing two of the triangles that make up the space between the points. Of the 7 groups mentioned above:
- 2 groups include none of these triangles (groups 1 and 6), so they remain groups of 5
- 4 groups include one of these triangles (groups 2, 3, 4, and 7), so they become groups of 3 instead
- 1 group (group 5) includes two of these spaces, so that group becomes a group of 2
Thus, we have 2*5 + 4*3 + 1*2 = 24 triangles.
Interaction between inner and outer Pentragrams
Now, if we look at how the outer and inner stars interact, then we can add an additional 15 triangles (3 groups of 5). Basically, take group 5 of the inner pentagram and pair it with first one point of the outer star, and then the other. Then take all but group 5 of the inner pentagram and pair it with the point of the outer star.
The spaces between the points of the outer pentagram do not contribute to any additional stars.
Thus, we have 35 + 24 + 15 = 74 triangles based on the pentagrams alone.
Now lets add the vertical line. This splits the top point into two triangles, and the top space in the interior pentagram as well as bottom point of the interior pentagram. This adds 6 triangles directly, but now introduces more groupings. It also splits both the inner and outer pentagons inscribed inside each star in half. These shapes aren't triangles, but can contribute as we see later.
The newly created half of the top point can be grouped with the half of the space below it, as well as all the way to the new half of the bottom point. As well, it can be grouped with the newly created half of the pentagon. (3 more triangles per side means 6 more triangles).
The half of the top space of the interior pentagram that was split can only be extended down to the new half of the bottom point (1 more triangle per side means 2 more triangles).
The half of the bottom point of the interior triangle can be grouped with the adjacent space to make a right angle triangle, or with the new half inner pentagon, or with the half inner pentagon and non-adjacent point. Also, it can be grouped with the adjacent space and point and half inner pentagon. (4 more triangle per side means 8 more triangles).
So now we have 6 + 6 + 2 + 8 = 22 more triangles.
Finally, lets add the horizontal line.
It directly creates 9 more triangles since it
- makes a new triangle within both points of the outer star (2)
- divides both spaces between two points of the inner star into two triangles (4)
- makes 3 new triangles in the inner pentagon - 2 small that combine to form the third.
New Triangle in Outer Points
If we look at the newly created triangle within one of the outer star's points, it contributes to 5 new triangles. To see this, start with just this new triangle and then move the top right point to the right one vertex so that now it includes the newly created triangle made from the bottom of 2 above (1). Now move the bottom point out the middle (2). Then move the top point to the middle (3) and then again to just past the middle (4). Lastly, move the bottom point all the way to the tip of the outer point, and the top point out one more vertex (5). Since this happens on both sides, this is 10 new triangles.
Lastly, there is one more triangle that includes both of the these newly created triangles which extends to the bottom point of the inner triangle.
Thus, 11 new triangles are created using these.
Spaces between points of inner pentagram
Of the space that was divided in two, the lower half actually doesn't not contribute to any new triangles that we haven't already counted. The upper half, however, contributes to 2 more (one with the half of the top point of the outer star, and one that uses the newly created triangles in the inner pentagon). This happens on both sides, so there is 4 more triangles.
Again, another triangle is created using both of these upper spaces and the upper point of the other star.
Thus, 5 new triangles are created using these.
New triangles inside the inner pentagon
Of the three new triangles in the middle, the smaller ones don't contribute to any new triangles on their own, but the big one contributes to 2 more by pairing with the point on either side.
Horizontal Line Summary
So, we have 9 + 11 + 5 + 2 = 27 more triangles.
Add these all up and you get 74 + 22 + 27 = 123 - the same as the previous answer.
How long did I take on this? About an hour writing it all up, but not as long to find them all.