Finding a circle tangent to two given circles and x-axis

Question

Given 2 circles with known radii and positions.

What is the position of the pink circle if it should touch the previous 2 circles and the x-axis?

A little bit more info:

It is a special case for this one previous related problem. But I find it difficult to represent AD, DB or DC cathetus.
This also could be useful info probably related problem

https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/LCC.shtml — Intelligenti pauca, Apr 14 '18 at 12:04
In your picture the circles are tangent and one of them is centered at the origin: can we assume that as true? — Intelligenti pauca, Apr 23 '18 at 17:31
yes, but circle C should be tangent to the x-axis. In provided solution, m line is perpendicular to "my" x-axis. can you, please, shed some more light on this solution? — Mezk Erei, Apr 25 '18 at 18:06

score 0 · Accepted Answer · answered Apr 25 '18 at 19:48

0

Let $C=(x,y)$ be the center of the pink circle, of radius $r_C=y$. Then we have: $$ \cases{ x^2+y^2=(r_B+y)^2\\ x^2+(r_A+r_B-y)^2=(r_A+y)^2} $$ This system is easy to solve for $x$ and $y$.

answered Apr 25 '18 at 19:48

Intelligenti pauca

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yes. thank you! – Mezk Erei Apr 26 '18 at 12:00

Finding a circle tangent to two given circles and x-axis

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