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Circles $C_1$ and $C_2$ have equal radii and are tangent to that same line $L$. Circle $C_3$ is tangent to $C_1$ and $C_2$. $x$ is the distance between the between the centers of $C_1$ and $C_2$. Find the distance $h$, from the center of $C_3$ to line $L$, in terms of $x$ and the radii of the three circles.

Here is the figure.

this is how much I got:

Let $R_1$, $R_2$ and $R_3$ be the radii of circles $C_1$, $C_2$ and $C_3$ respectively with $R_1 = R_2 = R$, then $h = C_3O + R$

Now how should I go forward. I'm stuck here

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closed as off-topic by TheGeekGreek, Namaste, Daniel W. Farlow, Mick, Watson Feb 10 '17 at 19:11

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – TheGeekGreek, Namaste, Daniel W. Farlow, Mick, Watson
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ What are the parameters given. When only radii of $C_{1}$ and $C_{2}$ and $x$ are given, there are multiple radii of $C_{3}$ that one can have. $\endgroup$ – Jan Feb 9 '17 at 8:22
  • $\begingroup$ It looks like you have to assign variables to the radii of the circles. Please edit the question to show what you have gotten so far and where you got stuck. $\endgroup$ – David K Feb 9 '17 at 8:23
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    $\begingroup$ You only need to know that the length of a line connecting the centers of two tangent circles is equal to the sum of their radii. Then use Pythagoras theorem $\endgroup$ – polfosol Feb 9 '17 at 8:34
  • $\begingroup$ $h$ is $r_{12}$ to which you add the height of a triangle of base $x/2$ and hypothenuse $r_{12}+r_3$. $\endgroup$ – Yves Daoust Feb 9 '17 at 22:09
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How to solve it: Draw a triangle between the centers of the circles. One side is $x$, the other two sides are $R+R_3.$ Now find the height of this triangle. This height is the length from the center of the small circle to the intersection of the two straight lines. To obtain $h$, simply add $R$ to the found height.

Good luck!

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  • $\begingroup$ Note that the triangle drawn in this answer is isoceles, and its height is the same as the segment you've already identified, $C_3O$ (if $O$ is where I think it is). $\endgroup$ – David K Feb 10 '17 at 3:25

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