INVERSE IN THE CIRCUMCIRCLE

Let ABC be a triangle, O the circumcenter, P a point different to O and Q a point not lying on the OP line.
The radical axis of the circumcircles of ABC and OQP intersect the line OP at P'. P' is the inverse of P wrt circumcircle of ABC
APH, Comment in Romantics of Geometry problem

ARCHIVES

Old yahoo group Anopolis Anopolis

ENNEADIC LINES

Definition
Let T1, T2, T3 be three triangles with the property: The Nine Point Circle centers N1, N2, N3 of T1, T2, T3, resp. are collinear.
I call the line N1N2N3 enneadic line of T1, T2, T3
See an example of an enneadic line in geometry of triangle Euclid 6930

Conjecture
Let A1A2A3... An be a regular n-gon, n>4.  
There are three triangles of the n-gon having an enneadic line.

Problem
How many "different" enneadic lines are in a n-gon ?
(two enneadic lines are "different" if the 2 triads of the triangles are not symmetric)

Similarly we may ask for enneadic lines of 4 triangles.

PICTURES

Pentagon

Hexagon

Heptagon 1

Heptagon 2

Heptakaidecagon

Heptagon (4 triangles. Elias Hagos)

Mail Antreas P. Hatzipolakis

MORLEY RELATED TRIANGLE CENTERS

MORLEY RELATED TRIANGLE CENTERS
BY ANTREAS HATZIPOLAKIS and CHRIS VAN TIENHOVEN, PETER MOSES, CESAR LOZADA

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Antreas Hatzipolakis and Chris van Tienhoven

X(6120) =  1st HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6121) =  2nd HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6122) =  3rd HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6123) =  4th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6124) =  5th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6125) =  6th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(66087) = HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE CENTER
X(66089) = PERSPECTOR OF THESE TRIANGLES: 1st MORLEY AND HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE

Antreas Hatzipolakis and Peter Moses

X(65155) =  X(2)X(15857)∩X(357)X(8065)
X(65156) =  HATZIPOLAKIS-MOSES EQUILATERAL TRIANGLE CENTER
X(65157) =  X(2)X(3276)∩X(5)X(3280)
X(65158) =  X(357)X(5456)∩X(358)X(1136)
X(65377) = CENTER OF HATZIPOLAKIS-MOSES-MORLEY HYPERBOLA
X(65378) = PERSPECTOR OF HATZIPOLAKIS-MOSES-MORLEY HYPERBOLA
X(65379) = X(357)X(1136)∩X(10632)X(16871)
X(65380) = ANTICOMPLEMENT OF X(65377)
X(65381) = 98TH HATZIPOLAKIS-MOSES-EULER POINT
X(65382) = TRILINEAR POLE OF LINE {6, 65378}
X(66329) = 1st HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66330) = 2nd HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66331) = 3rd HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66332) = CENTER OF THE 1st MORLEY CONIC
X(66333) = PERSPECTOR OF THE 1st MORLEY CONIC
X(66365) = X(356)-CEVA CONJUGATE OF X(357)
X(66366) = X(3276)-CEVA CONJUGATE OF X(1136)
X(66367) = X(3277)-CEVA CONJUGATE OF X(1134)
X(66619) = X(3275)X(3602)∩X(3276)X(3606)

Antreas Hatzipolakis and César Lozada

X(66478) = CENTER OF THE 1st EULER-ROUSSEL EQUILATERAL TRIANGLE
X(66479) = CENTER OF THE 2nd EULER-ROUSSEL EQUILATERAL TRIANGLE
X(66480) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EULER-ROUSSEL AND 1st MORLEY
X(66481) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER-ROUSSEL AND 1st MORLEY

César Lozada

X(66180) = CENTER OF THE 2nd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE
X(66181) = CENTER OF THE 3rd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE
X(66182) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd MORLEY AND 2nd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE
X(66183) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd MORLEY AND 3rd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE