The Three McCay Circles,
the First and Second Brocard Triangle
... and Several Discoveries
by
Markus Heisss
Würzburg, Bavaria
2018/2019/2024
Last update: December 31, 2024
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."
The line OK is the diameter of the Brocard circle,
with O = circumcenter and K = symmedian point.
And since we were talking about the SECOND Brocard triangle,
for the sake of completeness, here is the FIRST
Brocard triangle:
Now there is an interesting relationship between these two Brocard triangles:
Further informationen on the Brocard triangles can be found [here].
Now to the construction of the McCay circles:
In short: The symmetry axes of the Steiner ellipse
intersected with the respective bisectors of the triangle sides
provide the diameters of the three McCay circles.
Further information on the Steiner ellipse and its construction can be found [here].
The parameters of the McCay circles can also be calculated relatively easily:
The intersection points of the symmetry axes
with the bisectors of the triangle
sides (as shown in Fig. 04) are collinear.
Notes on 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 ABD, BCE and CAF are similar.
3.) So it has only to be prooved, that the points D, E, F and G are collinear.
The problem is shown in the following drawing:
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Further collinearities:
Nia, Nib, Nic, Noa, Nob, and Noc are the vertices of the Napoleon triangles.
More on that? [here]
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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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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:
