The Circle of Apollonius

 and Four New Circles

 

 by

Markus Heisss

Würzburg, Bavaria 

 2018/2020/2022

 

The copying of the following graphics is allowed, but without changes.

To get a bigger picture, please click it with the cursor.

 

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Firstly it has to be said, that there are different "Circles of Apollonius".

 In the following context we use the circle, which has the definition:

 All points Px, for which the ratio APx to BPx is constant, lie on a circle.

 

Apollonius circle, circle of Apollonius
Fig. 01: Circle of Apollonius

 

And now with values:

 

Apollonius, circle, theorem, formula, proof

 

[Further information is available on the internet under:

wikipedia.org ==> "Circle of Apollonius"]

 

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Postscript from April 17,  2022:

 

The circle of Apollonius with two orthogonal circles:

(All necessary information you'll find in the following figures.)

 

two orthogonal vertical circles, Apollonius
Fig. 01c: Circle of Apollonius with two orthogonal circles
 

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  Now to the four new circles:

 

Draw circles at points Px of the Appolonius circle with radii APx (or BPx).

They intersect the line BPx (or APx) at points, which again lie on a circle.

 

There are exactly four possibilities, shown in the following figures:

 

Appolonius of Perga, Geometry, Published by Markus Heiss, Würzburg, Bavaria
Fig. 02: Circle of Apollonius and a new circle, 1. possibility
Published by Markus Heiss, Würzburg, Bavaria
Fig. 03: Circle of Apollonius and a new circle, 2. possibility
Geometry, by Heisss
Fig. 04: Circle of Apollonius and a New Circle, 3. Possibility
circles, geometry, by Markus Heiss, Germany
Fig. 05: Circle of Apollonius and a New Circle, 4. Possibility

And now all four possibilities:

by Heisss, Geometrician from Würzburg, Germany, 2018
Fig. 06: Circle of Apollonius and a New Circle (All 4 Possibilities)

 

The tangents from point B to the circle of Apollonius

are also tangent to two of the four new circles!

 

 

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Postscript from March 15,  2020:

 

Discoveries of further relationships:

(All necessary information you'll find in the following figures.)

 

Apollonius circle, Geometry
Fig. 07: Circle of Apollonius and Three Further Discoveries

 

And another discovery:

 

Science, Discovery, Geometry, Math, Germany, Würzburg
Fig. 08: Circle of Apollonios and a Fourth Discovery

 

  This has an application at the so-called "McCay circles".

More? [here]

 (There you can find the proofs, too.)

 

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Are you interested in my other geometrical discoveries?

[here]