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February 26th, 2017, 11:36 AM  #1 
Newbie Joined: Feb 2017 From: Netherlands Posts: 8 Thanks: 2 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 interm 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: $\displaystyle \sin^{1}(\frac{cmdn}{cm+dm}) * 2 = ∠s$ 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? Last edited by skipjack; February 26th, 2017 at 12:46 PM. 
February 26th, 2017, 02:09 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 16,936 Thanks: 1253 
I can't read the annotation in your diagram.

February 26th, 2017, 02:21 PM  #3 
Newbie Joined: Feb 2017 From: Netherlands Posts: 8 Thanks: 2 Math Focus: Trigonometry and complex numbers  Oh. I am very sorry. Here is a link to the uncompressed version: https://s32.postimg.org/5fp4srmsl/Proof.png 

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circles, tangents, touching 
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