# Difficult geometry problem to find length [closed]

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. 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

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

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• 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. – Jan Feb 9 '17 at 8:22
• 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. – David K Feb 9 '17 at 8:23
• 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 – polfosol Feb 9 '17 at 8:34
• $h$ is $r_{12}$ to which you add the height of a triangle of base $x/2$ and hypothenuse $r_{12}+r_3$. – Yves Daoust Feb 9 '17 at 22:09

## 1 Answer

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!

• 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). – David K Feb 10 '17 at 3:25