We break the lines into sets $A_i, B_i$, where parallel lines belong to the same set, and in the coordinate plane, lines in set $A_i$ have non-negative gradient and do not include infinity, lines in set $B_i$ have negative gradient and include infinity, and the lines in $A_i$ are perpendicular to $B_i$. Let $|A_i|=a_i$, $|B_i|=b_i$, then we have $\sum(a_i+b_i)=100$, and $|T|=\sum a_ib_i(100-a_i-b_i)$. Note that $100=(100-a_i-b_i)+(a_i+b_i)\ge 2\sqrt{(a_i+b_i)(100-a_i-b_i)}\implies (a_i+b_i)(100-a_i-b_i)\le 2500$. Now by AM-GM, \begin{align*} 2\sum a_ib_i(100-a_i-b_i) & \le \sum (a_i^2+b_i^2)(100-a_i-b_i) \\ &=\sum (a_i+b_i)^2 (100-a_i-b_i)-2a_ib_i(100-a_i-b_i) \\ &\le 2500\sum (a_i+b_i)-\sum 2a_ib_i(100-a_i-b_i) \end{align*}so $\sum a_ib_i(100-a_i-b_i)\le \sum \frac{2500\sum (a_i+b_i)}{4}=62500$, and equality can be achieved when there are $25$ lines parallel to the lines $x=0,y=0, y=x$ and $y=-x$.