Dear All My Friends,

I would like to add some remarks to this topic:

1). Clark Kimberling already put this property in his ETC, at X(4) Orthocenter item:

http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X4

2). I. F. Sharygin, Problems on Geometry. Plane geometry, Nauka, Moscow, 1982, page 46, problem 188:

On the lines AB and AC take points M and N respectively. To prove that common chord of two circles with diameters CM an BN pass through intersection point of altitudes of triangle ABC.

3). V. V. Prasolov, Problems in Plane Geometry, Part 1, Nauka, Moscow, 1986, page 64, problem 3.23 (b):

On sides BC, CA, AB of acute triangle ABC take any points A1, B1, C1. To prove that three common chords of pairs of circles with diameters AA1, BB1, CC1 pass through intersection point of altitudes of triangle ABC.

4). Recently we can read the problem in book Mathematical Olympiad Challenges of T. Andreescu, R. Gelca (Birkhäuser, 2004, 5th printing, 1.3.8 (p. 12)) as in:

http://www.cut-the-knot.org/Curriculum/Geometry/PHQCollinearity.shtml

Best regards,
Bui Quang Tuan

--- In Hyacinthos@yahoogroups.com, Floor van Lamoen <f.v.lamoen@...> wrote:
>
> Dear all,
>
> At the moment it seems that I only have time to do tiny problems. And
> probably this one is already known, but I was surprised by it:
>
> Let A'B'C' be a triangle inscribed in ABC (so A' lies on BC, etc.).
> Determine the radical center of the circles with diameters AA', BB' and
> CC' respectively.
>
> Kind regards,
> Floor.
>

Dear All My Friends,

There is another interesting reference for this fact:

H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967, Theorems 2.45, 2.46

We can read detail at:

http://www.cut-the-knot.org/Curriculum/Geometry/CircleOnCevian.shtml

Best regards,
Bui Quang Tuan

--- In Hyacinthos@yahoogroups.com, "Quang Tuan" <bqtuan1962@...> wrote:
>
> Dear All My Friends,
>
> I would like to add some remarks to this topic:
>
> 1). Clark Kimberling already put this property in his ETC, at X(4) Orthocenter item:
>
> http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X4
>
> 2). I. F. Sharygin, Problems on Geometry. Plane geometry, Nauka, Moscow, 1982, page 46, problem 188:
>
> On the lines AB and AC take points M and N respectively. To prove that common chord of two circles with diameters CM an BN pass through intersection point of altitudes of triangle ABC.
>
> 3). V. V. Prasolov, Problems in Plane Geometry, Part 1, Nauka, Moscow, 1986, page 64, problem 3.23 (b):
>
> On sides BC, CA, AB of acute triangle ABC take any points A1, B1, C1. To prove that three common chords of pairs of circles with diameters AA1, BB1, CC1 pass through intersection point of altitudes of triangle ABC.
>
> 4). Recently we can read the problem in book Mathematical Olympiad Challenges of T. Andreescu, R. Gelca (Birkhäuser, 2004, 5th printing, 1.3.8 (p. 12)) as in:
>
> http://www.cut-the-knot.org/Curriculum/Geometry/PHQCollinearity.shtml
>
> Best regards,
> Bui Quang Tuan
>
> --- In Hyacinthos@yahoogroups.com, Floor van Lamoen <f.v.lamoen@> wrote:
> >
> > Dear all,
> >
> > At the moment it seems that I only have time to do tiny problems. And
> > probably this one is already known, but I was surprised by it:
> >
> > Let A'B'C' be a triangle inscribed in ABC (so A' lies on BC, etc.).
> > Determine the radical center of the circles with diameters AA', BB' and
> > CC' respectively.
> >
> > Kind regards,
> > Floor.
> >
>