- It also lies on line X155-X195, and is X(1147) of the Euler triangle.

--- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:

>

> It also lies at least lines X4-X110 and X5-X389.

>

> --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@> wrote:

> >

> > This point P, not in ETC, lies on line OQ with Q=X(1568) and satisfies the ratio OP:PQ=-OH^2/R^2.

> >

> > Francisco Javier.

> >

> > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:

> > >

> > > Let ABC be a triangle, A'B'C' the orthic triangle and

> > > A"B"C" the Euler triangle (ie A",B",C" are the second

> > > intersections of NPC with AA',BB',CC', resp. = midpoints

> > > of AH,BH,CH, resp.)

> > >

> > > Denote:

> > >

> > > Ra = the radical axis of ((B', B'C"),(C',C'B"))

> > >

> > > Rb = the radical axis of ((C', C'A"),(A',A'C"))

> > >

> > > Rc = the radical axis of ((A', A'B"),(B',B'A"))

> > >

> > > The Ra,Rb,Rc are concurrent.

> > >

> > > Point?

> > >

> > > Antreas

> > >

> >

> - Variations and Points

http://anthrakitis.blogspot.gr/2013/04/concurrent-radical-axes.html

APH

--- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:

>

> Let ABC be a triangle, A'B'C' the orthic triangle and

> A"B"C" the Euler triangle (ie A",B",C" are the second

> intersections of NPC with AA',BB',CC', resp. = midpoints

> of AH,BH,CH, resp.)

>

> Denote:

>

> Ra = the radical axis of ((B', B'C"),(C',C'B"))

>

> Rb = the radical axis of ((C', C'A"),(A',A'C"))

>

> Rc = the radical axis of ((A', A'B"),(B',B'A"))

>

> The Ra,Rb,Rc are concurrent.

>

> Point?

>

> Antreas

> - Antreas,

Your point P1.2 (Search = 2.3145425702586469385) is also the complement of X(1147) and the centroid of ABCX(68).

Randy

--- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:

>

> Variations and Points

>

> http://anthrakitis.blogspot.gr/2013/04/concurrent-radical-axes.html

>

> APH

>

> --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:

> >

> > Let ABC be a triangle, A'B'C' the orthic triangle and

> > A"B"C" the Euler triangle (ie A",B",C" are the second

> > intersections of NPC with AA',BB',CC', resp. = midpoints

> > of AH,BH,CH, resp.)

> >

> > Denote:

> >

> > Ra = the radical axis of ((B', B'C"),(C',C'B"))

> >

> > Rb = the radical axis of ((C', C'A"),(A',A'C"))

> >

> > Rc = the radical axis of ((A', A'B"),(B',B'A"))

> >

> > The Ra,Rb,Rc are concurrent.

> >

> > Point?

> >

> > Antreas

> >

> - APHThe entire plane?Which is the locus of P such that R1, R2, R3 are concurrent?Similarly R2, R3R1 = the radical axis of (ABaC), (ACaB)Ba, Ca = the orthogonal projections of B, C on AP, resp.Let ABC be a triangle and P a point.Denote:
[APH]:

Let ABC be a triangle and P a point.

Denote:

Ba, Ca = the orthogonal projections of B, C on AP, resp.

R1 = the radical axis of (ABaC), (ACaB)

Similarly R2, R3

Which is the locus of P such that R1, R2, R3 are concurrent?

The entire plane?

[César Lozada]:> Which is the locus of P such that R1, R2, R3 are concurrent? The entire plane?

Yes. For P=u:v:w (trilinears) they concur at Z(P)=u^2*cos(A) : :

ETC pairs(P,Z(P)):

(1,1), (2,75), (3,255), (4,158), (5,1087), (6,31), (7,1088), (8,341), (9,200), (10,1089), (11,1090), (12,1091), (15,1094), (16,1095), (19,1096), (20,1097), (21,1098), (30,1099), (31,560), (32,1917), (37,756), (40,1103), (42,872), (44,678), (46,1079), (55,1253), (56,1106), (57,269), (58,849), (63,326), (65,1254), (73,7138), (75,561), (76,1928), (81,757), (84,1256), (86,873), (88,679), (90,7042), (100,765), (101,1110), (110,1101), (174,7), (188,8), (190,7035), (192,8026), (238,8300), (259,55), (266,56), (365,6), (366,2), (483,179), (507,174), (508,85), (509,57), (513,244), (514,1111), (518,4712), (519,4738), (523,1109), (556,3596), (649,3248), (650,2310), (651,7045), (652,2638), (656,2632), (661,2643), (758,4736), (798,4117), (1049,1085), (1077,1028), (1125,6533), (1488,7002), (2089,7022), (2238,4094), (2292,6042), (3082,400), (4146,6063), (4166,220), (4179,594), (4182,346), (4367,7207), (6724,12), (6725,6057), (6726,480), (6727,60), (6728,11), (6729,3271), (6730,4081), (6731,5423), (6733,59), (7025,188), (7039,7044), (7041,7036), (7370,7023), (7371,479), (7591,7066), (9326,2226)

For Fermat points:

Z(X(13))= cos(A)*csc(A+Pi/3)^2 : : (trilinears)

= on cubics K420b, K638 and these lines: (5,8919), (13,15), (470,8838)

= [ -0.003162528585600, -0.00276260060789, 3.644036680137045 ]

Z(X(14))= cos(A)*csc(A-Pi/3)^2 : : (trilinears)

= on cubics K420a, K638 and these lines: (5,8918), (14,16), (471,8836), (5619,6774)

= [ 2.242952306040436, 4.57457502811627, -0.561557755730165 ]