Calculate miter points of stroked vectors in Cartesian plane
Draw two vectors $\vec u$ and $\vec v$ as in the picture below. They form a parallelogram having $a/2$ and $b/2$ as altitudes. It follows that $$ u={b\over 2\sin\beta},\quad v={a\over 2\sin\beta}. $$ It is then easy to compute $\vec u$ and $\vec v$: $$ \vec u={A-B\over AB}{b\over 2\sin\beta},\quad \vec v={C-B\over BC}{a\over 2\sin\beta} $$ and finally: $$ D=B+\vec u+\vec v,\quad E=B-\vec u-\vec v. $$
