Dear Darij

>[APH] Let Ra, Rb, Rc be the orthogonal

>projections of H on the cevians of I [ie

>on the angle bisectors of A,B,C, resp.]

>In my acute-angled triangle figure, the Euler lines

>of BCRa, CARb, ABRc are concurrent.

[DG]:

>Good idea... but unfortunately, zooming in on my

>dynamic geometry sketch shows that they are not

>exactly concurrent...

Hmmmm...... Apo idees .... allo tipota!!

[free translation: I am full of ideas!!]

Here is another one (idea, I mean):

What point is (where is lying on?) the Radical Center of the

circumcircles of the triangles bounded by the lines:

(AB,AC, parallel - to - BC - through - Ra)

(BC,BA, parallel - to - CA - through - Rb)

(CA,CB, parallel - to - AB - through - Rc) ?

Now, let Ta, Tb, Tc be three points on the angle bisectors

AI, BI, CI such that:

ITa / IRa = ITb / IRb = ITc / IRc = t

As t varies, which is the locus of the Radical Center of the

circumcircles of the triangles bounded by the lines:

(AB,AC, parallel - to - BC - through - Ta)

(BC,BA, parallel - to - CA - through - Tb)

(CA,CB, parallel - to - AB - through - TC) ?

Greetings

Antreas

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