1877. A Property Of Rectangular Circumhyperbola

1876. Metric Relations

1875. Same Centers

 1. ) Let  and  be the orthocenter and Nagel's point of . Let  be the projections of  on .

*  and  have same incenter.

2.) Let  be the symmedian point of . Let  be the projections of  on perpendicular bisectors of 
**  and  have same centroid.

1874. A Family Of Conics Centered at X(5)-NPC Center (Part 2)

1873. A Family Of Conics Centered at X(5)-NPC center

 Lat ABC be an acute angled triangle and P be a point.

A1B1C1-cevian triangle of P and A2B2C2-circumcevian triangle of P.
Oba=Circumcenter of (BA1A2),  Oca=Circumcenter of (CA1A2).
Circle through the points A2,Oba, Oca intersects the BC at A3, A4.
Define B3,B4,C3,C4 cyclically.

*A3, A4, B3, B4, C3, C4 lie on same conic. For all P, center of conic is X(5)-NPC center.
Are there any remarkable properties of perspectors of these family of conics?

1872. Concyclic Points

1871. Collinear Points

 Let I=X(1)- Incenter of ABC. P be a point on Circumcircle of ABC with first barycentric (a^2(x-y)(x-z):...:...)  Pa,Pb,Pc inverse of P in the circles BIC, CIA, AIB resp.

* I, Pa, Pb, Pc lie on same line. 
** Trilinear pole of  line has first barycentric (a (x-y)(x-z):...:...).
Some pairs (P,Q)=(Xi, Xj) where Q is trilinear pole of the line.
{(74, 2349), (98,1821), (99, 799), (100,190)....}

1870. Concyclic Points

1869. A Locus For K001-Neuberg Cubic (Part 4)

1868. Concyclic Points

1867. A Locus for K060 and Q161: K001-K060 half-way quartic

1866. A Property Of Kiepert Circum Hyperbola

 This problem is inspired from #5175 by Tran Quang Hung.

Let P,Q be two points on Kiepert hyperbola of ABC. Circles APQ, BPQ, CPQ intersects the Kiepert hyperbola of ABC at A', B',C' resp.
R=X(2)-Centroid of of A'B'C' lies on Kiepert hyperbola of ABC.
So we have a transformation T, T(P,Q)->R, Let's show it with * operation. Then we have:
P*Q=R, P*R=Q, Q*R=P

1865. A Locus For K001-Neuberg Cubic (Part 3)

1864. Perspective Cyclic Quadrilaretals

1863. Concyclic Points

1862. An Extension Of Van Lamoen Circle