# Radius and Center of the biggest possible circle $c_2$ inside circle $c_1$ without containing point $p$

giving a circle $c_1$ with radius $R_1$, Center$(X_1,Y_1)$ and a point $p$. (R1,X1,Y1 and p are known). I want to calculate $r_2$ and center point $x_2$,$y_2$ for circle $c_2$. $c_2$ covers the biggest possible area inside $c_1$ without containing point $p$ inside it.

• Please improve your drawing. – user535339 Mar 8 '18 at 17:20
• Align C2 so it has a diameter with P at one end and tangent to C1 at the other, maybe? – Oscar Lanzi Mar 8 '18 at 17:21
• @oscarLanzi i want to calculate r2,x2,y2 given r1,x1,y1 WITHOUT drawing (i sensed my question wasn't clear so i edited it) – kk96kk Mar 8 '18 at 17:40
• @idk done, plz edit the post thus it appears in it, I don't have privilege. – kk96kk Mar 8 '18 at 17:43

As Oscar Lanzi writes in his comment, the circle that you’re looking for will be tangent to the outer circle at one end of a diameter $\overline{pq}$. A tangent line to a circle is perpendicular to the diameter at the point of tangency, so $(x_1,y_1)$ lies on $\overline{pq}$. The center of the inner circle $(x_2,y_2)$ is of course the midpoint of $\overline{pq}$, and $R_2$ is just half the length of $\overline{pq}$, i.e., $$R_2 = \frac12\left(\|(x_1,y_1)-p\|+R_1\right)$$ and $$(x_2,y_2) = p + R_2{(x_1,y_1)-p \over \|(x_1,y_1)-p\|} = {R_2 \over 2R_2-R_1}(x_1,y_1) + {R_2-R_1 \over 2R_2-R_1}p.$$