Common Tangents

of Two Circles

 

The Discovery: Equal Chords

 

1.) In the first picture you see two circles,

each with the internal or external tangents.

(Sometimes they are called "inner and outer tangents".)

common tangents of two circles, external, internal, inner and outer, equal chords,

If you draw a line through the other tangent points as shown,

then the chords have the same lengths.

2.) The external tangents with formulas:

external tangents of two circles, outer tangents, excircles, formula, geometry, discoveries, Markus Heisss, Würzburg

The formula for the length of the two chords is:

... or as:   s1 = s2 = 4*R*r/d*((((d - R + r)(d + R - r))/(d*d + 4*R*r))^(1/2))


Further formulas:  


A short proof you will find here:

https://gogeometry.blogspot.com/2018/08/geometry-problem-1379-common-external.html

3.) And now the internal tangents with formulas:

internal Tangents, two circles, inner tangents, Markus Heiss, Heisss, Geometry

The formula for the length of the two chords is:

... or as:   s3 = s4 = 4*R*r/d*((((d + R + r)(d - R - r))/(d*d - 4*R*r))^(1/2))


Further formulas:  


A short proof you will find here:

https://gogeometry.blogspot.com/2018/08/geometry-problem-1380-common-internal.html

 

[Supplement:]

tangents of two ellipses, common, external, internal

4.) Another phenomenon you can see in the next picture:

internal tangents of two circles, or inner tangents, Markus Heiss or Heisss, Würzburg

Conjecture: If you draw lines through the new intersections

of the chords with the circles, you get four further chords,

which have the same length. You can do this again and again.

Furthermore you get intersections of the chords,

and the fragments of the chords have also the same lengths.

==>  The segments with the same colour in the drawing have the same lengths.

 

[This is also valid for two similar ellipses with one common axis of symmetry.]

 5.) If you rotate the figure with the internal (or external) tangents,

 then the chords cut off parts of spheres, which look like "apple peels".

 These apple peels have the same volumes.

Markus Heiss Heisss Würzburg

Be careful, if the chords are crossing the axis of rotation!

This is the formula for the volumes with internal tangents.


This is the formula for the volumes with external tangents.


6.) And the last picture:

Markus Heiss or Heisss, Würzburg, internal tangents of two circles, inner tangents, apple peels

 Conjecture: According to picture 4 you can rotate the figure

and get fragments of spheres, which have the same volumes.

 

7.) The whole geometrical phenomenon

was discovered in 2003 by Heisss.

 

I hope you enjoyed it.

 

References:

1.) Magazine: "Die Wurzel - Zeitschrift für Mathematik", Dec. 2005, p. 267  ==>  www.wurzel.org

2.) Website:  ==>  https://markus-heisss.jimdofree.com/geometrie-handskizzen/

 

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Are you interested in my other geometrical discoveries?

[here]