- Let A1A2A3A4A5A6 be a regular hexagon and P a point.

Name the triangles PA1A2, PA2A3, etc as 1,2,3,4,5,6.

The Euler lines of the even triangles (ie 2,4,6)

are concurrent at a point Q and of the odd triangles

(ie 1,3,5) at a different point R.

In General:

Let A1A2A3....An, be a regular n-gon, with n =3k,

and P a point. Name the triangles PA1A2, PA2A3, etc

as 1,2,3,....n.

Whose triangles the Euler lines are concurrent?

APH - One more:

Let ABC be an equilateral triangle, P a point, A;B'C' the

cevian triangle of P, and A",B",C" the reflections of

A,B,C in A',B',C', resp.

The Euler lines of AB"C", BC"A", CA"B"

are concurrent.

APH

--- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:

>

> Another problem:

>

> Let ABC, A'B'C' be two arbitrary equilateral triangles.

>

> Denote:

>

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

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

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

>

> The Euler lines of AAbAc, BBcBa, CCaCb are concurrent.

>

> Corollary:

>

> Let ABC be an equilateral triangle and P a point.

>

> Denote:

>

> Ab, Ac = the reflections of A in BP, CP, resp.

> Bc, Ba = the reflections of B in CP, AP, resp.

> Ca, Cb = the reflections of C in AP, BP, resp.

>

> The Euler lines of AAbAc, BBcBa, CCaCb are concurrent.

>

> Proofs?

>

> APH

>

> --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:

> >

> > Let ABC be an equilateral triangle and P a point.

> >

> > Which same central lines of the triangles PBC, PCA, PAB

> > are concurrent for all P's ?

> >

> > The OH (Euler) lines? The OI lines? ......

> >

> > APH

> >

>

---In Hyacinthos@yahoogroups.com, <anopolis72@...> wrote :One more:

Let ABC be an equilateral triangle, P a point, A;B'C' the

cevian triangle of P, and A",B",C" the reflections of

A,B,C in A',B',C', resp.

The Euler lines of AB"C", BC"A", CA"B"

are concurrent.

APH

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In general, they are not.

Which is the locus of P such that the Euler lines of AB"C", BC"A", CA"B" or

the Euler lines of A"BC, B"CA, C"AB are concurrent?

APH