HYACINTHOS 6531

Let ABC be a triangle and P an arbitrary point. Let A'B'C' be the
cevian triangle and A"B"C" the circumcevian triangle of P, and let
XYZ be the desmic mate of the triangles A'B'C' and A"B"C", i. e.

X = B'C" /\ B"C',
Y = C'A" /\ C"A',
Z = A'B" /\ A"B'.

Then,

(a) the lines AX, BY, CZ concur at a point Q;
(b) the lines A'X, B'Y, C'Z concur at a point Q';
(c) the lines A"X, B"Y, C"Z concur at a point Q".

While (b) and (c) are corollaries of the Desmic Theorem, (a) cannot
be proven in this way. Remark that if P has trilinears P(x:y:z), then
for Q, Q', Q", we get the trilinears

/ a b c \
Q ( -------- : -------- : -------- );
\ x(bz+cy) y(cx+az) z(ay+bx) /

Q' ( ax ((b²z²+c²y²)x + xyzbc + ayz(bz+cy)) : ... );

Q" ( ax ((b²z²+c²y²)x + ayz(bz+cy)) : ... ).

I call Q, Q', Q" the first, second and third umecevian points of P.
(Please don't ask me where the name comes from: I have thought it up,
using the German "Umkreis" and "ceva", and it has not much sense, but
a name is urgently needed.)

Note that the first umecevian point Q is the isogonal of the
complement of the isogonal of P.

A list of triangle centers and their umecevian points follows:

---- P = INCENTER X(1) ----

Point P( 1 : 1 : 1 ) = incenter X(1).
First umecevian point
Q( a/(b+c) : ... ) = isogonal Spieker point X(58).
Second umecevian point
Q'( a(b²+c² + bc+ca+ab) : ... ) = X(386),
intersection of Brocard axis (3,6) and Nagel line
(1,2).
Third umecevian point
Q"( a(b²+c² + ca+ab) : ... ), not in ETC.

---- P = CENTROID X(2) ----

Point P( 1/a : 1/b : 1/c ) = centroid X(2).
First umecevian point
Q( a/(b+c) : ... ) = Sigur symmedian point X(251).
Second umecevian point
Q', not in ETC.
Third umecevian point
Q", not in ETC.

---- P = CIRCUMCENTER X(3) ----

Point P( cos A : cos B : cos C ) = circumcenter X(3).
First umecevian point
Q( sec A : ... ) = orthocenter X(4).
Second umecevian point
Q', not in ETC.
Third umecevian point
Q"( (cos A)[1 - cos A cos(B-C)] : ... ) =
X(185), Nagel point of orthic triangle.

---- P = ORTHOCENTER X(4) ----

Point P( sec A : sec B : sec C ) = orthocenter X(4).
First umecevian point
Q( sec (B-C) : ... ) = Kosnita point X(54).
Second umecevian point
Q', not in ETC.
Third umecevian point
Q"( cos A - cos 2A cos(B-C) : ... ) = Taylor
center X(389). Can somebody find a synthetic
proof?

---- P = NINE-POINT CENTER X(5) ----

Point P( cos (B-C) : cos (C-A) : cos (A-B) ) =
nine-point center X(5).
All three umecevian points are not in ETC.

---- P = SYMMEDIAN POINT X(6) ----

Point P( a : b : c ) = symmedian point X(6).
All three umecevian points Q, Q', Q" coincide
with P = symmedian point X(6). This means that
X lies on AA', etc..

---- P = GERGONNE POINT X(7) ----

Point P( 1/(a(s-a)) : ... ) = Gergonne point X(7).
All three umecevian points are not in ETC.

---- P = NAGEL POINT X(8) ----

Point P( (s-a)/a : ... ) = Nagel point X(8).
All three umecevian points are not in ETC.

---- P = MITTEN POINT X(9) ----

Point P( s-a : s-b : s-c ) = Mitten point X(9).
All three umecevian points are not in ETC.

---- P = SPIEKER POINT X(10) ----

Point P( (b+c)/a : (c+a)/b : (a+b)/c ) =
Spieker point X(10).
All three umecevian points are not in ETC.

---- P = CLAWSON POINT X(19) ----

Point P( tan A : tan B : tan C ) = Clawson point X(19).
First umecevian point
Q( (sin A)/(cos B + cos C) : ... ) = X(284).
Second umecevian point
Q', not in ETC.
Third umecevian point
Q", not in ETC.

---- P = DE LONGCHAMPS POINT X(20) ----

Point P( cos A - cos B cos C ) = de Longchamps
point X(20).
All three umecevian points are not in ETC.

---- P = SCHIFFLER POINT X(21) ----

Point P( (s-a)/(b+c) : ... ) = Schiffler
point X(21).
First umecevian point
Q = X(961).
Second umecevian point
Q', not in ETC.
Third umecevian point
Q", not in ETC.

I am getting tired... Of course, the first umecevian point is
interesting if the point is the isogonal of an anticomplement of
something interesting -- try X(64), for example --. And on the other
hand, the second and the third umecevian points are so complicated
that I don't think that there are more cases where P and Q' are in
ETC.

Something more interesting: If P transverses a line, what curve make
Q, Q' and Q" ??

Sincerely,
Darij Grinberg