Finding solutions to intersecting circles, one with known radius and the other none.

Question

There are two circles, $C$ of radius $1$ and $C_r$ of radius $r$ which intersect on a plain. At each of the two intersecting points on the circumferences of $C$ and $C_r$, the tangent to $C$ and that to $C_r$ form an angle of $120$ degree outside of $C$ and $C_r$.

What formula do I use to find the distance between the centers of the two circles in terms of $r$?

Thanks @CYKwong for the help. – cartmanbrahhhhh May 15 '17 at 06:34 — cartmanbrahhhhh, May 15 '17 at 06:34

Robert Z · Accepted Answer · 2017-05-15T06:29:00.287

0

Note that the angles between the radii at the intersection point is $360-180-120=60$ degree i. e. $\pi/3$ radians (each radius is orthogonal to the corresponding tangent). Then distance $d$ between the centers is given by the Law of cosines: $$d^2=r^2+1^2-2r\cos(\pi/3)\implies d=\sqrt{r^2-r+1}.$$

edited May 15 '17 at 06:29

answered May 15 '17 at 06:23

Robert Z

145,942

Thanks man! I drew it out after reading your answer and found it. – cartmanbrahhhhh May 15 '17 at 06:35

Finding solutions to intersecting circles, one with known radius and the other none.

1 Answers1