- --

Let A'B'C' be the orthic and A"B"C" the medial

triangles of ABC.

Denote:

Ab, Ac = the reflections of A" in CC', BB', resp.

Bc, Ba = the reflections of B" in AA', CC', resp.

Ca, Cb = the reflections of C" in BB', AA', resp.

CONJECTURE:

The Nine Point Circles of the triangles

A"AbAc, B"BcBa, C"CaCb concur.

HAPPY NEW YEAR 2004 !

Greetings from Athens

Antreas - Dear Antreas
> Let ABC be a triangle, and A'B'C' its Orthic Triangle.

In both cases, your assertions are true and the locus of the common

>

> 1. Let Pa, Pb, Pc be points on AH, BH, CH, resp. such that:

>

> APa / AH = BPb / BH = CPc / CH = t

>

> Denote

>

> Ab, Ac = the reflections of Pa in CC', BB', resp.

> Bc, Ba = the reflections of Pb in AA', CC', resp.

> Ca, Cb = the reflections of Pc in BB', AA', resp.

>

> The Nine Point Circles of the Triangles

> PaAbAc, PbBcBa, PcCaCb are concurrent (??)

>

> Which is the locus of the point P of concurrence,

> as t varies?

>

> [If t = 0 (ie Pa = A, Pb = B, Pc = C), then P = X(1986) (BW)]

>

> 2. Let Pa, Pb, Pc be points on A'H, B'H, C'H, resp. such that:

>

> A'Pa / A'H = B'Pb / B'H = C'Pc / C'H = t'

>

> Denote

>

> Ab, Ac = the reflections of Pa in CC', BB', resp.

> Bc, Ba = the reflections of Pb in AA', CC', resp.

> Ca, Cb = the reflections of Pc in BB', AA', resp.

>

> The Nine Point Circles of the Triangles

> PaAbAc, PbBcBa, PcCaCb are concurrent (??)

>

> Which is the locus of the point P of concurrence,

> as t' varies?

>

> [If t' = 0 (ie Pa = A', Pb = B', Pc = C'), then P = X(1112) (JPE) ]

point of the three NP-circles is the line joining H to X(143) = the

NP-center of the orthic triangle

Friendly. Jean-Pierre - Dear Antreas
> Let A'B'C' be the orthic and A"B"C" the medial

They go through X(974)

> triangles of ABC.

>

> Denote:

>

> Ab, Ac = the reflections of A" in CC', BB', resp.

> Bc, Ba = the reflections of B" in AA', CC', resp.

> Ca, Cb = the reflections of C" in BB', AA', resp.

>

> CONJECTURE:

>

> The Nine Point Circles of the triangles

> A"AbAc, B"BcBa, C"CaCb concur.

Happy and peaceful 2004 to every Hyacinthist

Friendly. Jean-Pierre