- Thank you Bernard

You are right.

The three intersections are very close.

For the triangle ABC

with sin(A/2) = 3/5 sin(B/2) = 5/13

I found with MathCad for the three lines OI

a rational determinant with

nominator and denominator integer numbers

with about 20 digits and the value of the

fraction about 0.00005.

Best regards

Nik

--- In Hyacinthos@y..., Bernard Gibert <b.gibert@f...> wrote:

> Dear Antreas, Paul and Nik,

>

> >>> No surprise if the OI lines [after the OG and OK lines]

> >>> of the pedal triangles of Ia,Ib,Ic

> >>> are concurrent on some point on the Euler line of ABC as well!

> >>>

> >>> ***

> >>> A quick sketch suggest that they are.

>

> Cabri says they are NOT !

> although the three intersections are very close.

>

> >>>It would be interesting to know

> >>> the

> >>> point of concurrency.

> >

> [PY]

> > Now I have no confidence that the three lines are concurrent.

> > May be Edward can help check this.

> >

>

> Best regards

>

> Bernard - Dear Antreas

> Dear Gilles,

I apologize, you are right.. I am mistaked in my notations.

>

> [APH]

> Let IaaIabIac be the pedal triangle of Ia.

>

> A' := IaIaa /\ IabIac

>

> Similarly B', C'

>

> The triangles ABC, A'B'C' are perspective.

>

> [...]

>

>

> ==> the point of concurrence is the centroid G.

>

> [...]

>

> 1. A" := the orth. projection of Iaa on IabIac.

>

> Similarly B", C"

>

> Then the triangles ABC, A"B"C" are perspective.

>

> [GB]:

> The triangle MaMbMc with sides IabIac, IbaIbc, IcaIcb is homothetic to

> IaIbIc. A' is the feet on IabIac of the altitude, as we prove with

> Nikolaos Dergiades.

> A'B'C' is the orthic triangle of MaMbMc, and ABC is the orthic triangle

> of IaIbIc. So ABC and A'B'C' are homothetic, therefore they are

> perspective. Perspector is the Mittelpunkt X_9 of ABC.

>

> *********

>

> I guess your A'B'C' is mine A"B"C".

Gilles

>

>

> Let's see what the perspector of ABC, A"B"C" is:

>

>

> A

> ..................

> / \

> / \

> B----A1-Iaa--------C

> / \

> A3 A2

> Iac \

> / A" Iab

> / \

> Ia

>

> [...]

>

> ==> AA" is the a-cevian of Mittenpunkt (cot(A/2) ::)

>

>

> Antreas