HYACINTHOS 6315

In my Sharygin Points Report, message #6293, I wrote

>> §6. THE INTOUCH TRIANGLE AND S8
>>
>> Another interesting point is the homothetic center of the
>> triangle DEF and the intouch triangle of ABC. I call it the
>> 8TH SHARYGIN POINT of triangle ABC. It is not (yet) in ETC.
>>
>> NOTES: 1. The intouch triangle is the triangle whose vertices
>> are the points of tangency of the incircle with the sides of
>> triangle ABC. It is also called the Gergonne triangle of ABC.

>> It would be nice if somebody finds the trilinears of S8, S9,
>> S14 or S15.

The trilinears of S8 turned out to be remarkably simple:

/ b + c c + a a + b \
( ----- (aa - bc) : ----- (bb - ca) : ----- (cc - ab) ).
\ s - a s - b s - c /

Darij Grinberg