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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

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