HYACINTHOS 7225

I wrote:

>> I have just noted that if Ma, Mb, Mc are the midpoints
>> of BaCa, CbAb, AcBc, respectively, the lines AMa, BMb
>> and CMc concur at X(393), the barycentric square of
>> the orthocenter.

More is true. If X, Y, Z are the midpoints of AbAc, BcBa,
CaCb, respectively, then it is well-known that the lines
AX, BY, CZ concur at the Longchamps point X(20) of
triangle ABC, but it is less known that the lines MaX,
MbY, McZ concur at a point whose barycentrics are

( a^4 ( b^8+c^8-a^8 - 2a²(b^6+c^6)
+ 2b²c²(b^4+c^4-5a^4 + 3a²b²+3c²a²-3b²c²)
+ 2a^6(b²+c²) )
: ... )

after a Maple torture. The point is not in ETC.

Sincerely,
Darij Grinberg