My Math Forum Touching circles and tangents

 Trigonometry Trigonometry Math Forum

February 26th, 2017, 10:36 AM   #1
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Math Focus: Trigonometry and complex numbers
Touching circles and tangents

Hi, this is my first post on My Math Forum, so I apologize if I make any mistakes regarding syntax etc.

I have been playing with Thales' theorem and tangents for a bit and wanted to find the angle of the intersection facing two touching circles. (See attachment).

I first determined the radius of both circles, which in this particular case was 5 and 3. That means $\displaystyle cm = 5$ and $\displaystyle dn = 3$. That in-term means that $\displaystyle mn = 5 + 3 = 8$. I subtracted dn from cm, which equaled 2. That means gm = 2. Then I divided gm by mn which equaled $\displaystyle \frac{2}{8} = 0.25$.

I divided cm by 0.25, which equaled 20. That means $\displaystyle ms = 20$.

So using $\displaystyle \sin^{-1}$ I was able to calculate that $\displaystyle \sin^{-1}(\frac{cm}{ms}) = \sin^{-1}(\frac{5}{20}) = 14.4775... = ∠f$

This means $\displaystyle ∠s = 2*14.4775... = 28.9550...$

When I convert all of this to a formula I get the following:

$\displaystyle \sin^{-1}(\frac{cm-dn}{cm+dm}) * 2 = ∠s$

So my question is: is this formula correct and, if so, has it already been proven that it is correct? And what is the name of this formula?
Attached Images
 Proof.jpg (9.1 KB, 6 views)

Last edited by skipjack; February 26th, 2017 at 11:46 AM.

 February 26th, 2017, 01:09 PM #2 Global Moderator   Joined: Dec 2006 Posts: 17,725 Thanks: 1359 I can't read the annotation in your diagram.
February 26th, 2017, 01:21 PM   #3
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Joined: Feb 2017
From: Netherlands

Posts: 8
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Math Focus: Trigonometry and complex numbers
Quote:
 Originally Posted by skipjack I can't read the annotation in your diagram.
Oh. I am very sorry. Here is a link to the uncompressed version: https://s32.postimg.org/5fp4srmsl/Proof.png

 Tags circles, tangents, touching

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