Here are some simple geometry problems I am unable to resolve to my satisfaction. I asked the question on Math Stack (https://math.stackexchange.com/questions/2713754/a-problem-in-elementary-combinatorial-space-geometry) but it has received no interest, so I ask here in a different format.
Let $\delta$ denote the configuration of two intersecting lines, in other words, a conic with a double point, in three dimensional projective space.
Let $\nu$ denote the condition that one of the lines in $\delta$ intersects some given line.
Let $\mu$ denote the condition that the plane of $\delta$ passes through a given point.
Let $\rho$ denote the condition that the point of intersection of $\delta$ lies on a given plane.
Prove: $$\delta\mu^2\nu^4\rho=17$$ $$\delta\mu\nu^4\rho^2=17$$
$$\delta\mu\nu^6=70$$ $$\delta\nu^6\rho=70$$
$$\delta\mu\nu^6\rho=50$$$$\delta\mu\nu^5\rho=50$$
The first and second are the most baffling to me. The third and fourth, I have an idea, but I wish I had a better one.
Regarding the first, it means that the number of $\delta$ whose plane passes through two given points( $\mu^2$) and thus passes through a fixed line, and such that the lines of $\delta$ intersect $4$ given lines is $17$.
This can be analyzed as follows
${\bf Case 1}$ One of the lines of $\delta$ intersects three of the given lines. Then it intersects, these three lines and the axis of $\mu^2$, as ${\bf there \ are \ two \ lines \ intersecting \ four \ given \ lines \ in \ space}$ and there are $\binom{4}{3}=4$ such choices there are $8$ lines. The second line is then uniquely determined.
${\bf Case 2}$ This is the real problem. Each line of $\delta$ intersects $2$ of the given lines, as there are $\frac{1}{2}\binom{4}{2}=3$ such partitions each must have $3$ solutions, provided $17$ is the right answer. How to obtain this last calculation is the real problem. Well I have further thoughts, but I'll wait to see if any interest in this question. Am I missing something obvious ?
