Saturday, September 14, 2024

1987. Circumcenter On Euler Line

 Let H=X(4)-Orthocenter of ABC. A1B1C1 cevian triangle of H. B1A, C1A are reflections of B1, C1 on A.

LA, line through B1A, C1A. define LB, LC cyclically.

A2B2C2 triangle bounded by lines LA, LB, LC.


* Circumcenter of  A2B2C2 lies on Euler line of ABC.

Thursday, February 1, 2024

1969.Conics Intersecting Cubic Curves At a Fixed Point

 Let P be a triangle center on K001-Neuberg Cubic of ABC.

Any line through P intersect the  K001-Neuberg Cubic at P1 and P2.

*As P1, P2 varies on K001-Neuberg Cubic, the conic {A,B,C,P1,P2} intersect the K001-Neuberg Cubic at a fixed point.

**If P=X(1)-Incenter of ABC, fixed point is X(7164).
    If P=X(3)-Circumcenter of ABC, fixed point is X(1138).
    If P=X(4)-Orthocenter of ABC, fixed point is X(8431).
    If P=X(13)-1st Fermat-Toricelli point of ABC, fixed point is X(8445).
    If P=X(399) of ABC, fixed point is X(4).  

Thursday, January 25, 2024

1965. A Type Of Circlecevian and Concyclic Points

1964. A Construction For X(18569)

 Let A1B1C1 reflection triangle of X(3)-circumcenter on the side lines of ABC.

Tangents to circumcircle of A1B1C1 at A1, B1, C1 form a triangle A2B2C2.*

*Circumcenter of A2B2C2 lies on Euler line of ABC. It's X(18569).

1963. A Triangle Inscribed In Nine-Point Circle

 Let A1B1Cmedian triangle of ABC.

A2: Inverse of X(3)-Circumcenter of ABC in the circle with diameter B1C1
Define B2, C2 cyclically.

*A2B2C2 is inscribed in the Nine-Point Circle of ABC.

Sunday, July 16, 2023

1955. A Construction For the Perspector Of 2nd Lemoine Circle

Let K=X(6)-Symmedian point of ABC. Antiparallel from K to BC intersect the AB, AC at Ac, Ab resp.
Define Ba, Bc, Ca, Cb cyclically.
KA: Antipode of K respect to circle (KBaCa). Define KB, Kcyclically.

* ABC and KAKB,KC  are perspective triangles. Perspector is X(3527) = ISOGONAL CONJUGATE OF X(631)=perspector of 2nd Lemoine circle

1987. Circumcenter On Euler Line

 Let H=X(4)-Orthocenter of ABC. A 1 B 1 C 1 cevian triangle of H. B1 A , C1 A are reflections of B1, C1 on A. L A , line through B1 A , C1...