There are three circles as following (which are selected from the Golden Ratio), I wonder how to intersect the paths like pic 2
Is there any way to make them precisely tangent to each other in Illustrator?
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No problem. First start thinking about the problem in a horizontal or vertical axis that way its easy to snap.
Then snap the circles by their vertices together by dragging with the white selection arrow from one vertex while the whole object is selected. Then rotate the objects about the centers.
Image 1: Time lapse of sequence.
In this solution only the smallest circle will be moved. The two bigger circles have given places.
Be sure that you have smart quides and snapping to points ON. Deactivate other snappings.
In the following image (1) blue and green circles are fixed, the red one will be placed.
Draw radiuses for blue and green circles. They are only helping to find the centerpoints easily. Then make a copy of the red circle. Move both red circles to tangential positions of blue and green circles. Use the direct selection tool and drag by holding the anchor points. They snap easily to other anchor points.
Draw 2 auxiliary circles (black) to the centers of blue and green circles. You can start from the centerpoint by holding Alt+Shift as you draw. The auxiliary circles must be drawn through the centerpoints of the red circles.
Move the red circles to the crossing points of the black circles. Before moving add anchor points to the crossings for easy snapping.
Remove auxiliary drawings. Both possible solutions inside the blue and green circles are visible.
why this works:
Let red, blue and green circles have radiuses rR, rB and rG. The only possible places for the red circle are declared as follows:
Distance to blue and green circles must be = rR. Thus distances to the centerpoints of blue and green circles must be rB-rR and rG-rR. The black circles have those distances as radiuses.
Are there more solutions?
There are several more. You could drag copies of red circle also to external tangential positions. In step 2 we used only internal positions. Accordingly you have 2 more possible black circles. That makes a bunch of new crossings where a copy of red circle could be placed.
The completed solution in our original scenario looks out like this:
You probably can see that in some other intial scenario still 2 more solutions could exist. Total theoretical maximum is 8. In our example there's not enough room for the red circle in the space inside the green, but outside the blue circle.