- Bernard Gibert wrote:

>2). this particular one is described and noted (Ko) in "Le cercle d'Euler"

Dear Bernard

>by Collet & Griso.

I have a question about a locus which probably is included in the book

above, a book which I heve not seen:

Who is the author of this problem ?

Through the vertices of a triangle ABC we draw three parallels, and three

other paerpendicular to previous ones. So there are formed three rectangles

ADBE, BSCZ, and CKAH, which have as their diagonals the sides AB, BC,

CA, of the triangle. Find the locus of the intersection point of the three

other diagonals ED, SZ, KH.

(word-by-word translation from Panakis (*))

E A K

B D S

Z H C

The locus is the Euler Circle (as we [=Greeks, following you French (**)],

call the nine-point-circle or Feuerbach circle).

The triangle ABC is inscribed in the rectangle EKCZ.

It would be interesting the particular case that the rectangle is a square.

Asterisks:

(*) Panakis = I. Panakis: 2500 Problems of Geometric Loci With Their

Solutions [in Greek]. Athens, ca 1965, p. 654, #582.

The book has a general bibliography but not references for each one problem.

(**): Cf Victor Thebault:

A triangle ABC is inscribed in a circle (O) with a fixed diameter D,

and a transversal D', which turns about a fixed point, cuts BC, CA, AB

in A1, B1, C1. Let A2 and A3, B2 and B3, C2 and C3 be the orthogonal

projections of A and A1, B and B1, C and C1 on D.

(a) Show that the circles with centers at the the midpoints of AA1, BB1,

CC1 and passing through A2 and A3, and B2 and and B3, C2 and C3 meet

in a fixed point on the Euler circle, that is, the nine-point circle of

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

the triangle.

(b) Find the locus of the second point of interesection of the three

circles.

(The American Mathematical Monthly 45(1938) 554 by V. Thebault)

Antreas - Dear Antreas,

This is exactly Adam Bliss' conjecture of last May! I wish I had seen

that problem before!!

Kind regards,

Floor.

Antreas P. Hatzipolakis wrote:

<...>>

<...>

> Through the vertices of a triangle ABC we draw three parallels, and three

> other paerpendicular to previous ones. So there are formed three rectangles

> ADBE, BSCZ, and CKAH, which have as their diagonals the sides AB, BC,

> CA, of the triangle. Find the locus of the intersection point of the three

> other diagonals ED, SZ, KH.

> (word-by-word translation from Panakis (*))

>

> E A K

>

> B D S

>

> Z H C

>

> The locus is the Euler Circle (as we [=Greeks, following you French (**)],

> call the nine-point-circle or Feuerbach circle).

>

> The triangle ABC is inscribed in the rectangle EKCZ.

> It would be interesting the particular case that the rectangle is a square.

>

> Asterisks:

> (*) Panakis = I. Panakis: 2500 Problems of Geometric Loci With Their

> Solutions [in Greek]. Athens, ca 1965, p. 654, #582.

> The book has a general bibliography but not references for each one problem.