In the attached image there are two intersecting circles of differing size with centers A (0,0) and B (4,-4) and with radii of 3 and 5, respectively. The distance between A and B is shown to be 5.66, and there are two further points C and D which lie on the line segment between A and B as well on the circumference of A and B. How would one find the coordinates of C and D given the coordinates of A and B and the radii of the circles? It is known that C is 3 units from A since the radius is 3, and D is 5 units from B since the radius is 5. Sorry if this is not the best way to phrase the question.
The end goal is to find the midpoint of the overlapping region of the circles, which I am trying to find by finding the midpoint between C and D. This methodology will hopefully be expanded for 3 intersecting circles to find the midpoints of the then 4 overlapping regions.
Best Answer
We have $A=(0,0)$ and $B=(4,-4)$. The parametric equation of the line between those two points is given by the points $$(x,y)=(0,0)+t(4,-4)=t(4,-4),$$
for $t\in[0,1]$. When $t=0$ we get point $A$, when $t=1$ we get point $B$.
Because $C$ is $3$ units from $A$ along the line of length $\sqrt{32}$ from $A$ to $B$, the appropriate parameter value to get point $C$ is $$t_C=\frac{3}{\sqrt{32}}.$$ Similarly, since $D$ is $5$ units from $B$ ($\sqrt{32}-5$ units from $A$) along the line, the appropriate parameter value to get point $D$ is $$t_D=\frac{\sqrt{32}-5}{\sqrt{32}}.$$
The midpoint (call it $E$) is found by taking the average of these two $t$:
$$E=\frac{\sqrt{32}-2}{2\sqrt{32}}(4,-4)\approx (1.29,-1.29)$$