Draw a Circle Tangent to Three Given Circles

Tangent Circles

In an earlier sketch, I tackled a archetype problem of Apollonius: Construct a circle tangent to three capricious circles. I was subsequently brash by an acquaintance, John Del Grande, that my solution was incomplete. A circumvolve may be seen as a signal or a line, these being the limiting cases every bit the radius approaches zero or infinity. Rather than use three circles, nosotros should be using whatsoever combination of three from points, lines, and circles.

Dr. Del Grande listed all of the ten combinations in a textbook, Mathematics 12, by J. J. Del Grande, G. F. D. Duff, and J. C. Egsgard (1965 W. J. Gage Limited).

The developers of The Geometer's Sketchpad take been working to make Sketchpad documents that tin can be viewed on a web browser. Thank you to their efforts, the ten images immediately below are dynamic geometry images. The independent objects are reddish. The constructed circles are blue. If an independent object is a circle, then it may be manipulated by moving its heart, or past moving a indicate on the circle, which controls the radius. If the independent object is a line, then it is controlled by two points on the line. Contained points may exist moved freely.

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Iii Circles At most, viii solutions.
Two Circles, One Line At nigh, eight solutions.
Two Circles, I Bespeak At nearly, four solutions.
One Circle, Two Lines At most, eight solutions.
One Circle, One Line, One Signal At about, 4 solutions.
One Circumvolve, 2 Points At most, two solutions.
Three Lines At almost, four solutions.
Two Lines, I Bespeak At most, 2 solutions.
One Line, Two Points At most, two solutions.
Three Points At near, ane solution.

Construction Notes

That last construction is simply a circle through three points. It is covered in most elementary geometry courses. Some of the other constructions are extremely complicated. If you doubtfulness that, then open the Sketchpad version of the Three Circles cartoon and execute the Prove All Hidden command. You volition see something similar to the picture at right. This is one instance in which it is hard to dispute the value of dynamic geometry software. With then many interdependent objects, I would never exist able to consummate this drawing using straightedge and compass. It would exist too difficult to distinguish betwixt the different objects, the precision would exist compromised with each step, and and so many construction marks could not be cleanly erased.

Inversion geometry is used in the more difficult constructions. If you have never studied it, or if yous need a refresher on the terminology, this link may be helpful: Inversion Geometry

This begins with my own construction. I later learned of several others, at to the lowest degree one of which is much more elegant. This one comes first, nevertheless, because it is my ain.

It would seem difficult to observe a circle that is tangent to 3 given circles, but if two of those circles were concentric, then it would not be then difficult at all. Given any two non-intersecting circles, it is possible to define an inversion such that the circles' images are concentric. Invert all three circles, construct tangent circles, and run the solutions through the same transformation. Their images will be tangent to the original circles.

Begin with three circles.	Define the inversion.	Capsize the circles.
Construct a solution.	Invert the solution.	Take a bow.

What if the original circles intersect? That makes it even easier. Two intersecting circles can be inverted so that their images are intersecting lines. That would simplify the three-circumvolve trouble into a trouble of two lines and 1 circle. Below is the same sequence of steps, get-go with intersecting circles.

The Gergonne Solution

Observe that the construction to a higher place is actually two constructions. The one to use depends on whether any of the given circles intersect. That would not accept been such an issue in the 3rd century BC, but we at present live with dynamic geometry software, and the given conditions can modify even later the construction is complete. Information technology would be better to have one solution that fits all cases.

The construction below was published by Joseph-Diaz Gergonne in 1816. Information technology as well makes use of inversion geometry. I great reward is that the same construction works for all but a few special configurations of the given circles. For 3 given circles, the Gergonne construction will render all solutions and volition not result in any rogue circles, which are not solutions. This makes it work especially well with dynamic geometry. Exercise the construction one time, and it volition hold together while the organisation of the given circles is changed.

Proof of the Gergonne solution tin exist plant in the paper below. It is a journal commodity discussing solutions past François Viète, Isaac Newton, and Gergonne.

'The tangency trouble of Apollonius: 3 looks',
BSHM Message: Journal of the British Society for the History of Mathematics, 22:1, 34 - 46

The Gergonne construction:

The dilation point or eye of homothety of ii circles is the point from which 1 of the circles may be dilated onto the other. For two circles, there are generally two dilation points.

From a set of 3 circles, 3 different pairs can be taken. For each pair, construct the two dilation points. That results in six points. Through these six points, information technology is possible to draw four lines, each containing three of the dilation points. 3 of the lines separate one circle center from the other ii centers. The fourth line has all three center points on the same side.

Each of these lines leads to as many every bit two solutions. For clarity, only one of the lines and two of the solutions volition exist presented here. Construct the poles of the line with respect to each of the 3 circles.
Construct the radical centre of the three circles, and from that point, draw a line to each of the three poles. In this case, each line intersects its respective circle at two points.
The six intersection points are points of tangency for 2 solution circles, with three tangent points on each solution. The solutions can be synthetic from these points of tangency. Correctly separating them into groups of three may by a bit tricky. Here is i tip. If the line of similitude separates the middle of ane given circumvolve from the centers of the other two, and so the solution circle volition do also. In this sketch, the center of the lower red circle is separated from the others by the line of similitude. Therefore, one solution circle is internally tangent to this odd circle and externally tangent to the others. The 2d solution, vice verse.

Geometer'due south Sketchpad File

Beneath is a Geometer'southward Sketchpad document with constructed solutions for each of the ten cases. Most of the constructions are based on Gergonnes's solution.

Apollonius10.gsp

Dorsum to Whistler Alley Mathematics

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Last update: September 25, 2015 ... Paul Kunkel whistling@whistleralley.com
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Source: http://whistleralley.com/tangents/tangents.htm

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