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,

$\Large AP = \frac{R}{R + r} \times AB$
$\Large PB = \frac{r}{R+r} \times AB$
$\Large BQ = \frac{r}{R-r} \times AB$
$\Large AQ = \frac{R}{R-r} \times AB$

Now to my question,

Why did this formula work? I just analysis a pattern.
Have you guys ever heard of this formula before?

#Geometry

Easy Math Editor

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Comments

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 - 5 years ago

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

Rishabh Tiwari - 5 years ago

Ohhh, i get it

Jason Chrysoprase - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$