1988. Circumcenters On Euler Line

 Let ABC be a triangle with P, Q are two points on Euler line of ABC.

A',B',C' are inverse images of A,B,C through circle diameter PQ.

* Circumcenter of  A'B'C' lies on Euler line of ABC.

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.

1986. Circumcenters On Euler Line

1985. A Locus For K001-Neuberg Cubic Curve

 

1984. Congruent Circles

1983. A Circumcircle Through Circumcenter of ABC

1982. Collinear Points

 

1981. Euler To Euler

 

1980. Triangles Sharing The Same Centroid

1979. A Circle Through Midpoint

1978. An Inspiration Of Antreas Hatzipolakis's Problem

1977. Triangles With Same Centroid

1976. A Locus For K005-Napoleon-Feuerbach Cubic

1975. Isogonal Inscribed Triangles

 

1974. Concurrency Points On Euler Line

1973. A Construction For X(10020) = 22nd HATZIPOLAKIS-MONTESDEOCA POINT

1972. Collinear Points Related To K001-Neuberg Cubic

1971. Collinear Points On K001-Neuberg Cubic of ABC

1970. Lines Through Incenter

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).  

1968. Concyclic Points

1967. Concyclic Points

1966. A Lemoine Type Circle Centered At X(17508)

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 A1B1C1 median 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.

1962. A Type of Bicevian Conic

1961. Similar Triangles

1960. Triangles Inscribed In a Conic of ABC

1959. Circumcenter On Euler Line

1958. A Circumcenter On Euler Line of ABC

1957. Concurrent Circles On Circumcircle

1956. An Equilateral Triangle

 

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, KC cyclically.

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

1954. A Point On Median

1953. Concurrent Circles

1952. Concyclic Points

1951. Another Circle Centered At Incenter

1950. A Circle Centered At Incenter

1949. On Fermat's Configuration

1948. Gergonne-Nagel Conic

1947. A Conic Problem

1946. A-Mixtilinear Hyperbola

1945. Conics Related To Euler Line

1944. Collinear Points

1943. Four Congruent Circles

1942. Right Triangles Inscribed In Feuerbach Hyperbola

1941. A Property Of Orthocentroidal Circle

1940. Equal Area

1939. A Locus For K007-Lucas Cubic