What is the shortest distance between two circles, the first having center (5,3) and radius 12 and the other with center (2,-1) and radius 6.
When I draw the circle, I can figure out that the shortest distance between the circles can be calculating by drawing a line through the center point of both of circles, but that is only visual understanding. Is there any theorem behind this to make sure that my visual understanding is correct?
On
(The following is an adaptation of this related answer to the case of interior circles).
Let $A, B$ be the centers of the large and small circles, respectively. Let $C,D$ be the intersections of $AB$ with the small and large circles, respectively.
Suppose points $C',D'$ existed on the small and large circles, respectively, so that $C'D' \lt CD$. Since $BC=BC'$ and $AD=AD'$ that would imply:
$$ AB+BC'+C'D' \lt AB + BC'+CD = AB + BC + CD = AD = AD' $$
But the shortest distance between two points is the straight line, so the length of the broken line $ABC'D'$ can be no smaller than that of segment $AD'\,$, with equality iff $C'=D$, $D'=D$.
Hint: Name center of circles as $o(a_1,b_1),o'(a_2,b_2)$ usually take $oo'=d=\sqrt{(a_1-a_2)^2+(b_1-b_2)^2}$ then you have two cirlcle with radius $R,R'$ according to your figure ,minimum distance between two circles is $R-(d+R')$