Home

Kissing Circles

So suppose you have 3 points in 2D space.

For each of these 3 points, a circle with it’s center at these points appears.

These 3 circles are not any other circles. They are tangent to each other, or they are kissing each other.

So given any 3 points, how do you find the radii of these 3 circles? Give a thought to this question. Solutions will come out tomorrow.

Solution:

Lets name the points ${p}_{1}$${p}_{2}$ and ${p}_{3}$, and the radii of each circle be ${r}_{1}$, , ${r}_{2}$ and ${r}_{3}$. Also, let the distance between ${p}_{a}$ and ${p}_{b}$ be ${D}_{a,b}$.

So now,

${D}_{1,2} = {r}_{1} + {r}_{2} \quad ---(1)$

${D}_{3,2} = {r}_{3} + {r}_{2} \quad ---(2)$

${D}_{1,3} = {r}_{3} + {r}_{1} \quad ---(3)$

From $(1)$, $(2)$ and $(3)$ we can find that:

${r}_{1}=\frac{{D}_{1,2}+{D}_{1,3}-{D}_{3,2}}{2}$

${r}_{2}=\frac{{D}_{1,2}+{D}_{3,2}-{D}_{1,3}}{2}$

${r}_{3}=\frac{{D}_{1,3}+{D}_{3,2}-{D}_{1,2}}{2}$

Hence solved.

https://www.desmos.com/calculator/jjm17kozmi