HYACINTHOS 6545

[Darij Grinberg]:

Let I be the incenter and Ia, Ib, Ic the excenters of a triangle ABC.
Let X, Y, Z be the points of tangency of the incircle with BC, CA,
AB, and X', Y', Z' be the points of tangency of the respective
excircles with BC, CA, AB. Further let W be the Bevan point X(40) of
triangle ABC, i. e. the intersection of the lines IaX', IbY', IcZ'
and the circumcenter of triangle IaIbIc.

Then,

1. The circles AIX, BIY, CIZ are coaxal. They intersect at I and at
another point, the Schröder point of triangle ABC. The common chord
is the OI line of ABC.

2. The circles AIX', BIY', CIZ' are coaxal. They intersect at I and
at a new point.

3. The circles AIaX', BIbY', CIcZ' are coaxal. They intersect at two
points. The common chord passes through I.

4. The circles AWX', BWY', CWZ' are coaxal. They intersect at W and
at another point. The common chord is the OI line of ABC.

Has somebody proofs for 2. and 4.? Trilinears?

PROOF OF 3.: The circle AIaX' has diameter IaA", where A" is the
intersection of the extrenal angle bisector of A with BC, since the
angles IaX'A" and IaAA" are both 90°. So we have to prove that the
circles having diameters IaA", IbB", IcC" are coaxal, and that I lies
on the radical axis. But from the Bodenmiller and Steiner theorems
[see Hyacinthos #6124, §4], applied to the complete quadrilateral by
the lines IbIcA", IcIaB", IaIbC" and A"B"C", this circles are coaxal,
and the orthocenter of triangle IaIbIc, i. e. the incenter I of ABC,
lies on the radical axis. Now it remains to prove that in our case,
the circles have common points (in fact, there could be also the case
that they don't have common points). But this is easy: the incenter I
lies on the chords AIa, BIb, CIc of each of these circles; therefore,
he is an inner points of all three circles, and they must have common
points.

Sincerely,
Darij Grinberg

 

[Jean-Pierre Ehrmann]

Dear Darij

If I didn't mistake, the point in 2. is trilinear
x = (b+c-2a)(2bc(b+c-a)-(a+b+c)SA)/(b+c-3a)
The chord in 3. is the line IK
The point in 4. is trilinear
x = (b+c-2a)/(b+c-a)

Friendly. Jean-Pierre Ehrmann