Dear Antreas,

[APH]

>

> A'B'C' := the cevian triangle of P = (x:y:z)

>

> Dab := the centroid of B'C'B

> Dac := the centroid of C'B'C

>

> Dbc := the centroid of C'A'C

> Dba := the centroid of A'C'A

>

> Dca := the centroid of A'B'A

> Dcb := the centroid of B'A'B

>

>

> D1 := the centroid of DabDbcDca

> D2 := the centroid of DbaDcbDac

>

> I think that D1 = D2

> If so, then let D1 = D2 := P'

>

> Coordinates of P'?

>

> We can also define Da, etc, analogously to Ga, etc.

>

I also think D1 = D2.

P' = (y + z) (5 x^2 + 3 x y + 3 x z + y z) : :

Da = midpoint[Dba, Dca]

Db = midpoint[Dcb, Dab]

Da = midpoint[Dac, Dbc]

Da' = midpoint[Dab, Dac]

Db' = midpoint[Dbc, Dba]

Da' = midpoint[Dca, Dcb]

Da'' = midpoint[Dcb, Dbc]

Db'' = midpoint[Dca, Dac]

Dc'' = midpoint[Dab, Dba]

P'is the centroid of DaDbDc, Da'Db'Dc' & Da''Db''Dc''

Da* = Dba, Dbc /\ Dca, Dcb

Db* = Dcb, Dca /\ Dab, Dac

Dc* = Dac, Dab /\ Dbc, Dba

Da*= Db*= Dc* = The centroid of the cevian of P.

= x (y + z) (2 x + y + z) : :

Some ETC points,

{1,1962},{2,2},{4,51},{7,354},{8,210},{20,154},

{144,165},{253,1853}

looks like

Dac, Dab, Da', Da* are colinear and parallel to BC.

Similiarily

Dbc, Dba, Db', Db* // to CA

Dca, Dcb, Dc', Dc* // to AB

Best regards

Peter.