I'm out of time these days, but had a quick look. I didn't check the eqns in detail, so can't comment on the method, but didn't see anything obviously wrong, except that "line" is in the plottools package, which wasn't loaded. I just tried regular solve, without the parametric option, but still lost connection with the kernel, which shouldn't happen. So my two tips would be:
Change the floating point numbers like 0.5 to exact 1/2 etc, since solve is for symbolic solutions (maybe solve would do this internally, but it is better to be sure and it is anyways good practice).
Rearrange the tangent equations to remove the denominators so a/b=c changes to a=b*c. Maple's solve is very good for polynomial systems, so that may work better.