# Did i just make a new formula?

I was drawing two tangent circle many as I could with integer radius. I notice there is some pattern, so i quickly observe and find out 4 formula for two tangent circle.

Assume that $$AC$$ and $$AD$$ is $$R$$ , and $$BF$$ and $$BE$$ is $$r$$. Now these are the formula that i have founded,

1. $$\Large AP = \frac{R}{R + r} \times AB$$

2. $$\Large PB = \frac{r}{R+r} \times AB$$

3. $$\Large BQ = \frac{r}{R-r} \times AB$$

4. $$\Large AQ = \frac{R}{R-r} \times AB$$

Now to my question,

1. Why did this formula work? I just analysis a pattern.

2. Have you guys ever heard of this formula before?

Note by Jason Chrysoprase
1 year, 10 months ago

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Triangles ADP and BFP are similar with ratio $$R : r$$. Triangles ACQ and BEQ are similar with ratio $$R : r$$. So $$AP : AB = AP : (AP + PB) = R : (R + r)$$ by the similarity, and likewise with the rest.

- 1 year, 10 months ago

Similarity of triangles must be used here as Ivan sir suggested, btw nice observation jason.!

- 1 year, 10 months ago

Ohhh, i get it

- 1 year, 10 months ago