HYACINTHOS 6535

 

 

In the chapter on poristic triangles, Section 34, of his book "The
Modern Geometry of the Triangle", William Gallatly proves that the
poristic locus of the Gergonne point of a triangle is a circle coaxal
with the circumcircle and the incircle. We can use this to define a
new remarkable point:

Let O be the circumcenter and I the incenter of triangle ABC. Define
a point m on the line OI by the ratio

Om 4(R+r)
-- = ------,
OI 4R + r

where R is the circumradius and r the inradius of ABC. Then, if G is
the Gergonne point of ABC, we have

r(R-2r)
mG = -------.
4R + r

Thus m is the center of the circle which is the poristic locus of the
Gergonne point G.

I propose to call m the Greenhill point of ABC (as the theorem is
attributed to George Greenhill in Gallatly's book).

Rather nice trilinears of m are

m ( cos A + 4 cos B + 4 cos C
: cos B + 4 cos C + 4 cos A
: cos C + 4 cos A + 4 cos B ).

The point m is not in ETC.

Sincerely,
Darij Grinberg