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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
6 months, 1 week 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. Ivan Koswara · 6 months, 1 week ago

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@Ivan Koswara Similarity of triangles must be used here as Ivan sir suggested, btw nice observation jason.! Rishabh Tiwari · 6 months, 1 week ago

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@Ivan Koswara Ohhh, i get it Jason Chrysoprase · 6 months, 1 week ago

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