Constructing a circle with center belonging to a line, tangent to another line and passing through a point

Question

Given: two nonparallel lines $a$ and $b$ and point $P$.

Construct: a circle whose center belongs to $a$, passes through $P$ and is tangent to $b$.

Answer 1

Thanks to @conditionalMethod for pointing me to the parabola idea. One of the answers in the linked question (this one) gave me idea how to solve the problem even more simply:

1. Extend $a$ and $b$ until they intersect in $O$.
2. Build helper circle with center $A$ anywhere on $a$ and tangent to $b$.
3. Draw line through $P$ and $O$, intersecting the helper circle in $B$ and $C$.
4. Draw line parallel to $AB$ through $P$, intersecting $a$ in $Q$. This is the center of the one solution.
5. Draw line parallel to $AC$ through $P$, intersecting $a$ in $R$. This is the center of the other solution.