Find exact tangent points of a circle between a plain and circle

Question

There is a circle with $60$ units diameter, which is exactly $35$ units far from a plain. I want to place another circle with $50$ units diameter between them. How can I define exact $A$ and $B$ tangent points as it is shown in the picture?

Answer 1

Draw a horizontal line from the midpoint of the lower circle, going to the left where it meets the line labelled "35". It hits this line 25 units up (since that's the radius of the right-hand circle). Call this intersection $S$.

From $S$ to the circle center above it is $40$ units. From one circle center to the other is $30 + 25 = 55$ units. Hence from $S$ to the lower circle center is a distance $q$ such that $$ q^2 + 40^2 = 55^2 \\ q \approx 37.7491722 $$ So the point $B$ is $37.7491722$ units to the right of the place where the "35" line meets the top of the rectangle.

The point $A$ is $30/55$ths of the way from the center of the top circle to the center of the bottom circle, so $A$ is approximately $\frac{30}{55}37.7491722 \approx 20.59$ to the right of the "35" line, and at height $25 + \frac{25}{55}40 \approx 43.18$ (above the top edge of the rectangle).

Answer 2

It's trapezoid $O_1O_2BK$, where $O_1O_2=55$, $O_2B=25$ and $O_1K=65$.

Thus, by the Pythagoras theorem $KB=\sqrt{55^2-(65-25)^2}$