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I stand looking at a wall on which there are drawn three identical circles in a row. I hold up a coin with diameter 2cm at an arm’s distance (80cm) from my eyes (10cm apart). If I look through my left eye, the coin exactly covers the right circle, and if I look through my right eye, the coin exactly covers the left circle. How far do I stand from the wall?

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    $\begingroup$ do the 3 circles touch? $\endgroup$ – sds Feb 1 '17 at 20:39
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This is the view from above:

enter image description here

  • $A$ and $B$ are your eyes, $AB=10$
  • $GH=2$ is the coin
  • $EC=CD=DF=d$ are the 3 circles ($d$ is the unknown diameter)
  • distance from point $E$ to line $AB$ is $x$ (this is what you want to find)
  • distance from point $G$ to line $AB$ is 80 - your arm length
  • distance from point $J$ to line $AB$ is $a$ - unknown

From triangle similarity $ADF \sim AGH$: $$\frac{2}{80}=\frac{d}{x}$$ Thus $$x=40d$$

From triangle similarity $JGH \sim JAB$: $$\frac{2}{10}=\frac{a-80}{a}$$ Thus $$a=100$$

From triangle similarity $JGH \sim JCD$: $$\frac{2}{a-80}=\frac{d}{x-a}$$ Thus $$3d=10$$ and $$x=133\frac{1}{3}$$

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