Given two line segments $ab$ and $cd$, I want to draw a circle tangent to both line segments and passing through points $c$ and $b$.
Primitive operations available to me are:
- Draw a line between two points.
- Draw a perpendicular line passing through a point.
- Draw a line at a particular angle passing through a point.
- Construct a circle with a given line segment as the diameter.
- Construct a circle given 3 points on the boundary.
- Construct a circle given a center point and a point on the boundary.
- Find the midpoint of a line segment.
- Find center of an (already drawn) circle.
I can draw the circle passing through $c$ and $b$ and tangent to either $ab$ (or $cd$) by finding the intersection of a perpendicular bisector of $bc$ and a line from $b$ ($c$) perpendicular to $ab$ ($cd$).
However, I'm having trouble finding a circle passing through the points tangent to both line segments. Since I'm interested in actually drawing the such circles, obviously simpler constructions are better.
Update:
Note that if there's a certain symmetry between the line segments, then the above construction works perfectly. Suppose we have a line segment $a'b$ instead of $ab$. Then we get the following:
Find the centre of a circle passing through a known point and tangential to two known lines is a somewhat related question.
Best Answer
If $x$ is the point where your two lines meet, then there is no such circle unless $bx = cx$. Which is why you're having trouble.