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:
$$\delta\mu^2\nu^4\rho=17$$
$$\delta\mu\nu^4\rho^2=17$$

$$\delta\mu\nu^6=70$$
$$\delta\nu^6\rho=70$$

$$\delta\mu\nu^5\rho=50$$
 
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 ?