Tangencies: Three Tangent Circles

Any three points can be the centers of three mutually tangent circles. To construct the circles, form a triangle from the three centers, bisect its angles (blue), and drop perpendiculars from the point where the bisectors meet to the three sides (green). The points where these perpendiculars cross the sides are the desired points of tangency. They are also the points where an inscribed circle (red) is tangent to the triangle; this circle has its center at the point where the six lines meet, and crosses the three tangent circles perpendicularly at their tangent points.

If you were running Java, you'd see a nice animation instead of this gif.

The same construction works to form two circles inside a third, but you should use two external bisectors and one internal bisector instead of three internal bisectors. (Curious fact: three bisectors meet iff the number of external ones is even.) If you move the anchor points in the animation around, you may be able to reach a configuration like this.

Similarly to this pattern, four tangent circles also always have a circle through their tangencies, but it is not always perpendicular to them.

Animation created by Cinderella.
From the Geometry Junkyard, computational and recreational geometry.
David Eppstein, Theory Group, ICS, UC Irvine.

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