Given a segment $R$, a segment $H$, and an angle $\angle ABC$, construct a circle whose center is on ray $BA$, whose radius is congruent to $R$, and where the chord created by ray $BC$ is congruent to $H$.
Source: Based on Kiselev's Geometry
I've found this construction to be surprising challenging. I've reduced it to the following construction, which I'm stuck on:
Given an angle $\angle ABC$ and a segment $P$, find a point on ray $BA$ such that the perpendicular from that point to ray $BC$ is congruent to $P$
Or even more simply:
Given two intersecting lines, find the point where their distance is $P$
How should I proceed from here? Can you help me take the next step: either with my reduced problem, or an entirely new approach to the original problem?
Basis for reduction: Let the constructed circle have center $O$, and let the chords endpoints on the circle be $U$ and $V$. Then $\triangle OUV$ is an isosceles triangle with sides $R, R, H$ and the altitude from $O$ to $UV$ is also a median and hence a perpendicular bisector of $UV$. We need to find a point on $BA$ where $O$ is, which we can find if we can solve the second problem.
I assume you want a straightedge-and-compass construction. You can make the reduced construction like this:
Draw a line perpendicular to $BC$ passing through $B$. Mark point $D$ on that perpendicular such that $\left| BD \right| = P$. Draw a line perpendicular to $BD$ through $D$, and find its intersection with $BA$. The intersection is the point you are looking for.