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[EMHL] Grebe triangle: results (was: Grebe triangle)
Posted:
Feb 15, 2003 2:01 PM


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 wellknown 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 Grebeorthic perspector, has homogeneous trilinears
/ (D+aÃÂ²)(2D+bÃÂ²+cÃÂ²) (D+bÃÂ²)(2D+cÃÂ²+aÃÂ²) (D+cÃÂ²)(2D+aÃÂ²+bÃÂ²) \ (  :  :  ), \ cos A cos B cos C /
where D is the area of triangle ABC.
By the way, for A' we have trilinears
A' ( 2D + bÃÂ² + cÃÂ² : ab : ca ),
or equivalently A' (  2D  bÃÂ²  cÃÂ² : 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 + aÃÂ² + bÃÂ² + cÃÂ²)/(2area) cos A  2 sin A : ... )
= ( (1 + 2 cot A + 2 cot B + 2 cot C) cos A  2 sin A : ... ).
Neither the outer Grebeorthic 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 Grebeorthic perspector and has trilinears
/ (DaÃÂ²)(2DbÃÂ²cÃÂ²) (DbÃÂ²)(2DcÃÂ²aÃÂ²) (DcÃÂ²)(2DaÃÂ²bÃÂ²) \ (  :  :  ). \ cos A cos B cos C /
The vertices of the inner Grebe triangle have trilinears
A" ( 2D  bÃÂ²  cÃÂ² : 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  aÃÂ²  bÃÂ²  cÃÂ²)/(2area) cos A + 2 sin A : ... )
= ( (1  2 cot A  2 cot B  2 cot C) cos A + 2 sin A : ... ).
Neither the inner Grebeorthic perspector, nor T' is (yet) in ETC.
 TWO POINTS WITHOUT GEOMETRICAL DESCRIPTION
We have identified the points with trilinears
/ (D+aÃÂ²)(2D+bÃÂ²+cÃÂ²) (D+bÃÂ²)(2D+cÃÂ²+aÃÂ²) (D+cÃÂ²)(2D+aÃÂ²+bÃÂ²) \ (  :  :  ) \ cos A cos B cos C /
and
/ (DaÃÂ²)(2DbÃÂ²cÃÂ²) (DbÃÂ²)(2DcÃÂ²aÃÂ²) (DcÃÂ²)(2DaÃÂ²bÃÂ²) \ (  :  :  ) \ cos A cos B cos C /
as the outer and inner Grebeorthic perspectors. But we can also define two other points,
/ (D+aÃÂ²)(2DbÃÂ²cÃÂ²) (D+bÃÂ²)(2DcÃÂ²aÃÂ²) (D+cÃÂ²)(2DaÃÂ²bÃÂ²) \ (  :  :  ) \ cos A cos B cos C /
and
/ (DaÃÂ²)(2D+bÃÂ²+cÃÂ²) (DbÃÂ²)(2D+cÃÂ²+aÃÂ²) (DcÃÂ²)(2D+aÃÂ²+bÃÂ²) \ (  :  :  ), \ 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



