Dear Jean-Pierre,

In Hyacinthos message #9148, you wrote:

>> [DG]

>> > Please help me with the following problem (I need it as

>> > a lemma):

>> >

>> > Given a cyclic quadrilateral ABCD with the circumcenter

>> > O. The perpendicular to BD through B meets the

>> > perpendicular to AC through C at E. The perpendicular

>> > to BD through D meets the perpendicular to AC through A

>> > at F. Finally, let X be the intersection of the lines

>> > AB and CD. Then, the points O, E, F, X are collinear.

>> >

>> > Well, it is easy to show that O is the midpoint of EF;

>> > hence, O, E, F are collinear, but how about X ?

>>

>> [JPE]

>> Let A' and D' be the reflections of A and D about O.

>> O = AA' inter DD', E = CA' inter BD' and X = AB inter DC

>> are collinear by Pascal's theorem.

Thanks for the proof!

And here is how I came up with the problem above:

Some time ago I solved the problem 3 in the 2nd Round

of the Bundeswettbewerb Mathematik (German National

Mathematics Competition) 2003. The problem was:

Given a cyclic quadrilateral ABCD, let S be the meet

of the diagonals AC and BD, and K and L the

orthogonal projections of S on the sides AB and CD.

Show that the perpendicular bisector of the segment

KL bisects the sides BC and DA.

Well, the proof is not very easy, but not too

complicated, too. However, I found an interesting

additional result: The circumcircles of triangles

SBC, SDA and SKL and the circle with diameter SO are

coaxal, i. e. they have a common point different from

S. Hereby, O means the circumcenter of our

quadrilateral ABCD.

Now, it is easy to show that the circumcircle of

triangle SBC has the segment SE as diameter, the

circumcircle of triangle SDA has the segment SF as

diameter, and the circumcircle of triangle SKL has

the segment SX as diameter. Finally, it remains to

show that the circles with diameters SE, SF, SX and

SO are coaxal.

This is very easy using the collinearity of E, F, X,

O you have proven.

Thanks again, and I will send a paper on the

Bundeswettbewerb problem, together with my solution,

the above result with its proof and some other

extensions to the little German mathematics

periodical "Die Wurzel" with the corresponding

credit to you in the proof of the above result. I

can send you a PDF file of the paper if you are

interested (it is in German, however).

Sincerely,

Darij Grinberg