Πέμπτη, 9 Μαΐου 2013

NPCs. SEQUENCE OF POINTS

Let ABC be a triangle and P a point.

Denote:

A1, B1, C1 = The NPC centers of PBC, PCA, PAB, resp.

r11,r12,r13 = the radical axes of the NPCs: ((B1),(C1)), ((C1),(A1)), ((A1),(B1)), resp.

R1 = the point of concurrence of r11,r12,r13 [Radical center of the circles. It is the Poncelet point of P wrt ABC, lying on the NPC of ABC]

f11, f12, f13 = the parallels to r11, r12, r13 through A,B,C, resp.

The lines f11, f12, f13 concur at a point F1.

F1 is the reflection of P in R1.

Locus:

As P moves on a line (the Euler line, for example) which is the locus of F1 ?

Sequences of Ri, Fi:

Denote:

A2,B2,C2 = the NPC centers of A1BC, B1CA, C1AB, resp.

r21,r22,r23 = the radical axes of the NPC: ((B2),(C2)), ((C2),(A2)), ((A2),(B2)), resp.

R2 = the point of concurrence of r21,r22,r23

f21, f22, f23 = the parallels to r21, r22, r23 through A,B,C, resp.

The lines f21, f22, f23 concur at a point F2.

Similarly R3,R4...., Rn and F3,F4,.... Fn.

Where are lying the points Ri, Fi ?

Special points P: P = N, I

Antreas P. Hatzipolakis, 9 May 2013