two lines and a circle construction

[maelkyzzydec]

Here is something that I came across while solving another problem. I do not know the answer to this question - yet
With only a compass and straight edge, Given two intersecting lines and a circle, construct a circle that is tangent to the three objects. If anyone could save me some time figuring this out I would be grateful.

[Guv]

I do not know how to solve your problem, but you should start by drawing the bisector of the angle between the two intersecting lines.

A circle tangent to both lines will be on that bisector.

Next, I would experimentally draw a good approximation to a solution. A little fooling around should get you a good diagram.

Analyzing the diagram might give you a clue.

[kedaman]

(X1-x1)^2+(Y1-y1)^2=r1^2
(X1-x2)^2+(Y1-y2)^2=r2^2
Y2-y3=k3(X2-x3)
(X2-x1)^2+(Y2-y1)^2=r1^2
Y3-y4=k4(X3-x4)
(X3-x1)^2+(Y3-y1)^2=r1^2
solve r1, x1 and y1. r2,x2,y2 is your arbitrary chosen circle, k3,x3,y3 and k4,x4,y4 your lines, k3!=k4.
further on you have the intersections X1,Y1 between the circles, X2,Y2 and X3,Y3 between the drawn circle and each line.

[kedaman]

hmm you need to have
(x1-x2)^2+(y1-xy)^2=(r1+r2)^2
k3=(x3-x1)/(y3-y1)
k4=(x4-x1)/(y4-y1)
as well