HYACINTHOS 6537

THIS MESSAGE CONTAINS HYACINTHOS #6445, CORRECTED AFTER HYACINTHOS
#6463, AND SOME NEW RESULTS.

We will treat two Grebe triangles of a triangle: the outer Grebe
triangle and the inner Grebe triangle.

--- THE OUTER GREBE TRIANGLE

Erect squares BBaCaC, CCbAbA, AAcBcB on the sides BC, CA, AB of
triangle ABC (outwards). Then, the triangle A'B'C' enclosed by the
lines BaCa, CbAb, AcBc is called the outer Grebe triangle of ABC. It
is perspective (and homothetic) with triangle ABC, the perspector
being the symmedian point of ABC (this is a well-known fact).

With the help of computer reducing, I have found out that the outer
Grebe triangle is also in perspective with the orthic triangle of
ABC. The perspector, which I call outer Grebe-orthic perspector, has
homogeneous trilinears

/ (D+a²)(2D+b²+c²) (D+b²)(2D+c²+a²) (D+c²)(2D+a²+b²) \
( ---------------- : ---------------- : ---------------- ),
\ cos A cos B cos C /

where D is the area of triangle ABC.

By the way, for A' we have trilinears

A' ( 2D + b² + c² : -ab : -ca ),

or equivalently A' ( - 2D - b² - c² : ab : ca ).

The circumcenter T of the outer Grebe triangle A'B'C' lies on the
Brocard axis of ABC. The trilinears of T are

( (1 + 2 cot w) cos A - 2 sin A
: (1 + 2 cot w) cos B - 2 sin B
: (1 + 2 cot w) cos C - 2 sin C ),

where w is the Brocard angle of triangle of triangle ABC.

(Note that 1 + 2 cot w is the homothetic factor of triangles ABC and
A'B'C': for example, B'C' = (1 + 2 cot w) a.)

We can also write the trilinears of T as

( (2area + a² + b² + c²)/(2area) cos A - 2 sin A : ... )

= ( (1 + 2 cot A + 2 cot B + 2 cot C) cos A - 2 sin A : ... ).

Neither the outer Grebe-orthic perspector, nor T is (yet) in ETC.

--- THE INNER GREBE TRIANGLE

Edward Brisse had pointed me to a modification of the outer Grebe
triangle:

The inner Grebe triangle results if we erect the squares BBaCaC,
CCbAbA, AAcBcB inwards instead of outwards. Then, we get the inner
Grebe triangle A"B"C" enclosed by the lines BaCa, CbAb, AcBc.

This inner Grebe triangle A"B"C" is also perspective to the orthic
triangle of ABC. The perspector is called the inner Grebe-orthic
perspector and has trilinears

/ (D-a²)(2D-b²-c²) (D-b²)(2D-c²-a²) (D-c²)(2D-a²-b²) \
( ---------------- : ---------------- : ---------------- ).
\ cos A cos B cos C /

The vertices of the inner Grebe triangle have trilinears

A" ( 2D - b² - c² : ab : ca ) etc..

The circumcenter T' of the inner Grebe triangle A"B"C" lies on the
Brocard axis of ABC. The trilinears of T' are

( (1 - 2 cot w) cos A + 2 sin A
: (1 - 2 cot w) cos B + 2 sin B
: (1 - 2 cot w) cos C + 2 sin C ),

where w is the Brocard angle of triangle of triangle ABC.

(Now 1 - 2 cot w is the homothetic factor of triangles ABC and
A"B"C": for example, B"C" = (1 - 2 cot w) a.)

We can also write the trilinears of T' as

( (2area - a² - b² - c²)/(2area) cos A + 2 sin A : ... )

= ( (1 - 2 cot A - 2 cot B - 2 cot C) cos A + 2 sin A : ... ).

Neither the inner Grebe-orthic perspector, nor T' is (yet) in ETC.

--- TWO POINTS WITHOUT GEOMETRICAL DESCRIPTION

We have identified the points with trilinears

/ (D+a²)(2D+b²+c²) (D+b²)(2D+c²+a²) (D+c²)(2D+a²+b²) \
( ---------------- : ---------------- : ---------------- )
\ cos A cos B cos C /

and

/ (D-a²)(2D-b²-c²) (D-b²)(2D-c²-a²) (D-c²)(2D-a²-b²) \
( ---------------- : ---------------- : ---------------- )
\ cos A cos B cos C /

as the outer and inner Grebe-orthic perspectors. But we can also
define two other points,

/ (D+a²)(2D-b²-c²) (D+b²)(2D-c²-a²) (D+c²)(2D-a²-b²) \
( ---------------- : ---------------- : ---------------- )
\ cos A cos B cos C /

and

/ (D-a²)(2D+b²+c²) (D-b²)(2D+c²+a²) (D-c²)(2D+a²+b²) \
( ---------------- : ---------------- : ---------------- ),
\ cos A cos B cos C /

which are also not in ETC. I don't know of a geometrical
signification of these points.

Darij Grinberg