HYACINTHOS 6536

As we have considered the intersections of the circumcircle and the
nine-point circle, two points that are real only for obtuse
triangles, let me mention two points real only for acute triangles.
These are the isologic points.

In Theorem 6 of Paul Yiu's note "The Volume of an Isosceles
Tetrahedron and the Euler Line", MIQ 11 (2001), p. 15-19,

http://www.math.fau.edu/yiu/MIQ2001tetrahedron.ps

it is shown that

There are two points L-+ in the plane of triangle ABC
whose distances from the vertices of triangle ABC are
proportional to the lengths of the opposite sides if
and only if ABC is acute-angled. These two points lie
on the Euler line, dividing HO [H = orthocenter and
O = circumcenter] in the ratio
HO : OL-+ = 1 : lambda-+
[where
/------
(1+4w)-+4 \/ w(1+w)
lambda-+ = ---------------------,
/----
\/ 1-8w
where w = cos A cos B cos C].
These values of lambda are real if and only if the
triangle is acute.

The points L-+ are called ISOLOGIC POINTS.

I am quite interested in their trilinears (although I don't expect
them to be very simple...), because they have some similarities with
the isodynamic points I-+, and I suppose that some collinearities are
possible (like the many collinearities related to the Fermats,
Napoleons, and Isodynamics).

Sincerely,
Darij Grinberg