I have two concentric circles, one smaller and one larger. I need to geometrically construct a circular sector that will touch the outer and inner circles at its ends. The conditions are as follows:
- At the outer end, the angle between the sector and the tangent to the circle should be 30 degrees.
- At the inner circle, the sector should be perpendicular (i.e., it should point towards the center of both circles).
In terms of construction, this involves constructing a circle X that is tangential (touches line L at point P) and is also perpendicular to circle C (touches an imaginary line M that passes through the center of circle C, but we don't know this line; the imaginary line M is not parallel to line L).
I'm struggling with how to approach this problem. Any guidance or suggestions would be greatly appreciated.
Let the given circles have common centre $A$ and radii $r=AD$, $R=AF$ (with $r<R$, see figure below).
We want to construct a third circle, with centre $E$ such that $DE\perp DA$ and radius $d=DE$, intersecting the outer circle at a point $F$, such that $\angle AFE=30°$. That is, we must find the value of $d$.
By the cosine law applied to triangle $AFE$ we get: $$ AE^2=EF^2+AF^2-2EF\cdot AF\cos30°, $$ that is: $$ d^2+r^2=d^2+R^2-dR\sqrt3. $$ From there one immediately gets $$ d={R^2-r^2\over \sqrt3\, R}. $$