Two Intersecting Circles, Third Circle Inside One Of Them

Number of solutions: 4

SOLUTION METHOD

Loci of points satisfying certain properties

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Steps

0. There are circles c, d and e, the circle c lies in the circle d.
1. Draw hyperbolas g and q, the centres of the circles lie on them and the circles touch circles c and e.
2. Draw ellipses p and t, the centres of the circles lie on them and the circles touch circles c a d.
3. Find points of intersection of hyperbolas and ellipses.
4. Draw lines which connect the points of intersection and the centres of the assigned circles.
5. Draw the final circles with centres at the intersections of hyperbolas and ellipses, they touch the circles in their intersections with the connecting lines from step 4.
6. the individual circles apply according to the mutual position of the circles

SOLUTION METHOD

Resizing, dilation

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Steps

0. Three circles k₁, k₂ and k₃ are given. Two of them intersect, the third lies inside both of them.
1. First, we will look for solutions that have internal contact with the circle k₃. We will solve the problem using dilation. We will imagine that the solution circles we are looking for are reduced so that they pass through the center of S₃. In order to preserve the contact between these circles and the circles k₁ and k₂, the circles k₁ and k₂ must shrink by the radius of the circle k₃.
2. By dilation, we have changed the kkk problem to a Bkk problem, where the point is the point S₃ and the circles are the images of the circles k₁ and k₂. We will solve this problem using circular inversion. We will use k₃ with center S₃ as the base circle (we can use any circle with center S₃, so why not use k₃).
3. The image of the dilated solutions are the common tangents of the circles "k₁ ''" and "k₂ ''".
4. We display the found tangents back in the circular inversion. The obtained circles are the dilated images of the searched solutions. The centers S₄ and S₇ are already the centers of the searched solutions.
5. We enlarge the images of circles k₄ and k₇ in the dilation by the radius of circle k₃. This gives the first two solutions to the problem.
6. We have found two solutions to the problem that have inner contact with the circle k₃. Now we repeat the procedure to find the solution with an outer tangent.
7. While in the previous case the circles k₁ and k₂ were getting smaller in dilation, this time they will be getting larger.
8. The dilated images of circles k₁ and k₂ are shown in circular inversion with the main circle k₃.
9. Find the common tangents of the circles shown.
10. Display the found tangents back in the circular inversion. Their images are also the dilated images of the solutions we are looking for.
11. Reduce the circles k₅ ' and k₆ ' by dilating them by the radius of the circle k₃. Thus we have obtained both solutions with external contact to the circle k₃.
12. In total, we found four solutions to the problem.