X(4240) = DAO TWELVE EULER LINES POINT
http://faculty.evansville.edu/ck6/encyclopedia/ETCPart3.html#X4240
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b4 + c4 - 2a4 + a2b2 + a2c2 - 2b2c2)/[(b2 - c2)(b2 + c2 - a2)]
X(4240) = 3X(2) - 4X(402) = 3X(2) - 2X(1650) = X(2) + 2X(3081) = 2X(402) - 3X(1651) = 2X(402) + 3X(3081) = X(1650) - 3X(1651) = X(1650) + 3X(3081) (Peter Moses, April 3, 2013)
As a point on the Euler line, X(4240) has Shinagawa coefficients ((E - 8F)F,6(E + F)F - 2S2).
X(4240) is the point of intersection of the Euler lines of twelve triangles, constructed as in the next few sentences.
Let E be the Euler line of a triangle ABC. Let A1 = E∩BC, and define B1 and C1 cyclically. Let AB be the reflection of A in B1, and define BC and CA cyclically. Let AC be the reflection of C in B1, and define BA and CB cyclically. The Euler lines of the four triangles ABC, AABAC, BBCBA, CCACB concur in X(4240). (Dao Thanh Oai, Problem 1 in attachment to ADGEOM #1709, September 15, 2014). See also Telv Cohl, 'Dao's Theorem on the Concurrence of Three Euler Lines,' International Journal of Geometry 3 (2014) 70-73: Dao's Theorem.
Continuing, let A*B*C* be the paralogic triangle of ABC whose perspectrix is E. Then X(4240) lies on the Euler line of A*B*C*. (Dao Thanh Oai, noted just after Figure 1 in attachment to ADGEOM #1709, September 15, 2014).
Continuing, redefine AB as the point on line AC and AC as the point on line AB such that B1, A1, AB, AC line on a circle and A1, AB, AC are collinear. Define BC and BA cyclically, and define CA and CB cyclically. Let A2 = BABC∩CACB and define B2 and C2 cyclically. The Euler lines of the five triangles ABC, A2B2C2, AABAC, BBCBA, CCACB concur in X(4240). (Dao Thanh Oai, Problem 2 in attachment to ADGEOM #1709, September 15, 2014).
Continuing, The Euler lines of the triangles A2BACA, B2CBAB, C2ACBC concur in X(4240). (Dao Thanh Oai, September 17, 2014).
X(4240) is the only point on the Euler line whose trilinear polar is parallel to the Euler line. (Randy Hutson, January 29, 2015)
X(4240) is one of three points used to define a 10-point circle, denoted by D at X(7740). (Dao Thanh Oai, June 24, 2015)
X(4240) lies on these lines:
{{2,3}, {107,110}, {112,1302}, {146,5667}, {476,1304}, {523,5502}, {685,5466}, {877,5468}, {925,1301}, {2407,3233}<
X(4240) = midpoint of X(1561) and X(3081)
X(4240) = reflection of X(i) in X(j) for these (i,j): (2,1651), (1650,402)
X(4240) = anticomplement of X(1650)
X(4240) = X(687)-Ceva conjugate of X(112)
X(4240) = X(i)-cross conjugate of X(j) for these (i,j): (1511, 250), (2420, 2407)
X(4240) = trilinear pole of line X(30)X(1990)
X(4240) = homothetic center of Gossard and anticomplementary triangles
X(4240) = {X(i),X(j)}harmonic conjugate of X(k) for these (i,j,k): (402,1650,2), (1650,1651,402), (2409,4230,4226)
X(4240) = X(i)-isoconjugate of X(j) for these (i,j): (48,2394), (63,2433), (74,656), (525,2159), (647,2349), (656,74), (810,1494), (1304,2632), (1494,810), (2159,525), (2349,647), (2394,48), (2433,63), (2632,1304)
