1
$\begingroup$

Here is a badly drawn MS Paint representation of the problem:

We have two circles, shown in red, each with radius $R$ and tangent to one another. A straight line, shown in purple, is drawn tangent to both red circles. We then draw a blue circle tangent to both red circles and the purple line. What is the radius of the blue circle, in terms of $R$?

$\endgroup$
1
$\begingroup$

Call the radius $r$. Then by Pythagorean Theorem $$R^2+(R-r)^2=(R+r)^2$$ so $$R^2=4rR\implies r={R\over 4}$$

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thank you! It felt like there was a simple Pythagorean Theorem argument lurking in there but I just couldn't see it. $\endgroup$ – Curtis H. Apr 25 '14 at 0:43
  • $\begingroup$ Care to elaborate on the answer? How is it that by the Pythagorean Theorem you arrive at $R^2+(R−r)^2=(R+r)^2$ ? $\endgroup$ – user168608 Aug 8 '14 at 5:30
  • $\begingroup$ @Angel Draw a right triangle with hypotenuse between the centers of a red circle and the blue circle, and with the point of intersection of the two red circles as its third vertex. $\endgroup$ – user142299 Aug 8 '14 at 23:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.