Consider two lines, written as $y=m_1x+b_1$ and $y=m_2x+b_2$; their $y$-intercepts must be between $-1$ and 1, inclusive ($-1\le b_1 \le 1$ and $-1\le b_2 \le 1$). Thus, their intersection may fall inside, outside, or on the unit circle, given by $x^2+y^2=1$.
The point of intersection of the two lines is $\left(\frac{b_{2}-b_{1}}{m_{1}-m_{2}},\ \frac{m_{1}b_{2}-m_{2}b_{1}}{m_{1}-m_{2}}\right)$.
For this point to be inside the unit circle, the following inequality needs to be true: $(b_2-b_1)^2 + (m_1b_2 - m_2b_1)^2 < (m_1-m_2)^2$.
If we square root both sides we have something that looks like the distance formula: $\sqrt{(b_1-b_2)^2 + (m_1b_2 - m_2b_1)^2} < |m_1-m_2|$.
Thus, the criteria for determining whether the intersection of these two lines will fall into the circle or not is if the distance between points $(b_1, m_1b_2)$ and $(b_2, m_2b_1)$ is less than the difference of the slopes. I'm wondering if there is anything significant about these two points in drawing some sort of meaningful conclusion, or if it is just something natural that coincidentally happened to look intriguing.
Thanks!