We have 3 kissing circles. The radius of the circles are 12 cm. The bottom 2 circles are resting on the flat floor. Look at the drawing. How high is the dotted line from the floor?
Let's use geogebra to answer the question.
1)Draw points (12,12) and (36,12), they are called A and B.
2) Use the circle with center and radius tool and choose A and B as the centers of 2 circles. Fill in the box with 12 when prompted.
You should have the bottom two circles of the drawing resting on the x axis ( floor ).
3) Zoom and stretch the axes until the circles look round.
4) Draw another point at (24,36) called C.
5) Draw a perpendicular line through C.
6) Hide point C. Place a point on the perpendicular line called D using the point on object tool.
7)Make a circle with radius 12.
8) Move D down until the top circle just touches the two lower ones. Use care and zoom in to get the best intersection you can.
9)Get the intersection of the top circle and the perpendicular line using the intersect tool. It will be called E.
10) Draw a horizontal line through E that is dotted.
11) Read off the y value of E, it is 20.8. That is the height of the dotted line we want.
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.