The Three McCay Circles
of a Triangle
... and Several New Discoveries
by
Markus Heisss
Würzburg, Bavaria
2018/2019
The copying of the following graphics is allowed, but without changes.
[To get a bigger picture, please click it with the cursor.]
The three McCay circles of a triangle are defined as
the three circumcircles through the centroid
and pairs of the vertices of the second Brocard triangle.
But the McCay circles are simpler derived from the so-called Neuberg circles:
Radius a-Neuberg circle RNa = ...
Radius a-McCay circle RCa = 1/3 RNa = (formula see next graphic)
Distance between center of the a-Neuberg circle to the side a: dNa = ...
Distance between center of the a-McCay circle to the side a: dCa = 1/3 dNa = (formula see next graphic)
Both centers lie on the perpendicular bisector of side a.
The same with b- and c-McCay circle.
[Further information you get from the internet under:
mathworld.wolfram.com ==> "Neuberg circles", "McCay circles" and "second Brocard triangle"]
The next graphic shows the intersections of the McCay circles with the respective perpendicular bisectors:
Postscript, August 5th 2019:
The further research showed,
that these both collinearities are identical
with the axes of symmetry of the Steiner circumellipse!
(Heisss, July 2019)
Application:
This circumstance allows a 'real' construction of the three McCay circles.
(More on that you'll find below.)
Further information to the Steiner circumellipse and its construction you get here:
https://steiner-circumellipse.jimdofree.com/
Notes for the proof of collinearity:
1.) The McCay circles and centroid G show three times Thales's Theorem.
If the respective points are collinear, then the two lines would be perpendicular in centroid G.
2.) The distances and radii of the McCay circles to the respective triangle sides are direct proportional.
Therefore the triangles ABMxc, BCMxa and CAMxb are similar.
3.) So it has only to be prooved, that the points Mxa, Mxb, Mxc and G are collinear.
The problem is shown in the following drawing:
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... and two further collinearities:
Now all four collinearities together:
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Postscript, August 7th 2019:
Here the research showed,
that there exists also a connection
to the Steiner circumellipse:
Transform the Steiner circumellipse into a circle,
then the angle between the lines with the collinear points
and the major semi-axis of the Steiner circumellipse
is equal to the angle between the transformed lines
and the minor semi-axis. (see grahic below)
(Heisss, August 2019)
The calculation of the angle sigma:
More about cosΦ and the formulas of the semi-axes you find here:
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Postscript, 26th October 2019:
... and so much more collinear points!
Further research brought to light,
that the two relationships shown above are only special cases,
because there are a lot more collinear points!
(see next graphic)
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Postscript, 30th October 2019:
... and another special case!
Draw the tangents to the McCay circles
from the respective midpoints of the triangle sides.
(Two of the McCay circles are not shown in the next graphic,
because we know, that we get a pair of collinear lines.)
We transform these collinear lines again as described in Fig. 07
and find a 45°-angle.
(see next graphic)
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Now a 'real' and simple construction
of the three McCay circles of a given triangle:
1) Construct the two axes of the Steiner circumellipse as shown in the figure below.
2) Construct the perpendiculas bisectors on each side of the triangle.
3) The intersections of these perpendiculas bisectors with the two axes of the Steiner circumellipse
are the diameters of the McCay circles.
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Postscript, 19th December 2019:
Further Relationships to the McCay circles:
And now the same again, but with more Apollonius circles:
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Postscript, 17th January 2020:
Relationship between McCay circles and Brocard angle:
[Further information under:
mathworld.wolfram.com ==> "Brocard angle" and "first Brocard triangle"]
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Postscript, 7th September 2020:
... and a few collinearities:
