1837. Orthocenters On Pedal Circles

1836. A Center On Pedal Circle of Fermat Point

1835. A Circle Through Circumcenter

1834. Inversion Related to Isodynamic Points

1833. Inscribed Triangle

1832. Inverted Triangles and Circumcenters On Euler Line

 Let ABC be a triangle and A1B1C1 is perspective with ABC at a point P and homothetic to ABC.

A2 is inversion of A1 respect to circumcircle of ABC. Define B2, C2 cyclically.

* Circumcenter of A2B2C2 lies on the line OP.

Application. Let P be a point on Euler line of ABC. A1,B1,C2 midpoints of AP, BP, CP resp.

A2 is inversion of A1 respect to circumcircle of ABC. Define B2, C2 cyclically.

** O',Circumcenter of A2B2C2 lies on the Euler Line of ABC.

For P=X(2), O'=X(33532)

For P=X(5), O'=X(?)

For P=X(20), O'=X(12084)

1831. ConConic Triangles

 Let P be a point and A1B1C1 cevian triangle of P.

ta= tangent to circumcircle of A1B1C1 at A1. Define tb, tc cyclically.
Let A2B2C2 triangle bounded by lines ta, tb, tc.

*A,B,C, A2,B2,C2 lie on same conic.
** For P=X(2), Center of the conic is X(15449)-CENTER OF THE X(2)-ALTINTAS HYPERBOLA.
*** Which is the center of the conic for some triangle centers?

1830. Concyclic Points

1829. Constructions For X(14790) and X(18569)

 1. Let O=X(3) circumcenter of ABC. Ha, Hb, Hc orthocenters of OBC, OCA, OAB resp.

La=Antiparallel from Ha to BC. Define Lb, Lc cyclically.
Let A1B1C1 triangle bounded by lines La,Lb,Lc.
* Orthocenter of A1B1C1 lies on Euler line of ABC. Its X(14790)= Conjugate-Couple of X(22) In the Range In Involution {X(2),X(5)}, {X(3),X(4)}.

2.Let H=X(4) orthocenter of ABC. Oa, Ob, Oc circumcenters of HBC, HCA, HAB resp.
La=Antiparallel from Oa to BC. Define Lb, Lc cyclically.
Let A1B1C1 triangle bounded by lines La,Lb,Lc.
* Circumcenter of A1B1C1 lies on Euler line of ABC. Its X(18569)= Ehrmann-Side-Ehrmann-Vertex Similarity Image of X(382).

1828. Concyclic Points, Euler Line

1827. Similar Triangles, Isogonal Conjugates

1826. Myakishev Associate Triangles Continued

1825. Another Myakishev Associate Triangles

1824. Orthocentroidal Circle, Congruent Circles, Translation

1823. Congruent Circles