Suppose you have an arc segment of known radius but unknown arc length. From the ends of this arc, two lines are constructed, perpendicular to one another, such that one line is vertical and the other horizontal. With this information, I want to find the length of the segment of the horizontal line that reaches from the end of the arc to the intersection with the vertical line. I've included the best attempt at a diagram I could put together on my phone below (I have no idea how big it will be):
In the diagram, for clarity, I want to find the length BD. I also have that BC:BD::2:3
(A is the centre of the circle)
My working so far has been to start by constructing a line CD to form the right triangle BCD. Since I know the ratios of the shorter sides, I can work out the other two angles in the triangle, which came out to be roughly 34 and 56 degrees. If I knew that BD passed through the centre, I would be able to subtract the arc length subtended by an angle of 34 degrees from the arc of half the circle to get the arc length CD. I could then use that to find the angle CAD, and then I'd have an icoseles triangle (AD=AC=r) where I know all the angles and two sides, and so can calculate the third side. Then, since I'd know the length of CD and I know the ratio of the lengths of the sides of BCD, I'd be done.
But BD doesn't pass through A. What I need to work out is if it's possible to move forward from here, to work out the length of BD with what is known.
If you want the story behind the problem, it's under this spoiler: