On 1-06-02, Antreas P. Hatzipolakis <

xpolakis@...> wrote:

>Let 1,2,3,4 be four lines forming the

>complete quadrilateral (1234).

>

>The Miquel point of (1234) is the common point

>of the circles (123), (234), (341), (412)

>[and also the circle (O(123)O(234)O(341)O(412))

>ie the circle passing through the four O's of the

>triangles (123), (234), (341), (412)]

>

>Now, define the Miquel point of (1234)

>by using only LINES; not circles.

The solution is based in two lemmata (well-known theorems):

Lemma #1 :

The orthocenters H(123), H(234), H(341), H(412)

of the four triangles (123), (234), (341), (412)

are collinear (Steiner Line of (1234)).

Lemma #2:

Let P be a point. The reflections of the line

HP in the sidelines of triangle ABC concur in

a point on the circumcircle of ABC.

==>

The Miquel Point of the complete quadrilateral

(1234) is the point of concurrence of the reflections

of the Steiner line of (1234) in its sidelines 1,2,3,4.

Antreas