A hexagon theorem

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• In the Greek mathematical periodical Diastasis [=dimension] #3-4, 1998, pp. 24-37, I read the article by Nikos Kyriazis: A New Geometry Theorem and its more
Message 1 of 3 , Jul 1, 2000
In the Greek mathematical periodical Diastasis [=dimension] #3-4, 1998,
pp. 24-37, I read the article by Nikos Kyriazis:
A New Geometry Theorem and its more Significant Applications.

The author uses the theorem as a lemma to prove well-known theorems:
The three angle-bisectors/medians/altitudes of a triangle concur, etc.

Actually the theorem is not new. The earliest reference I found is the
following:

Soit A_1A_2A_3A_4A_5A_6 un hexagone convexe inscrit dans un cercle.
Si le produit A_1A_2 X A_3A_4 X A_5A_6 des cotes de rang pair egale
le produit des cotes de rang impair A_2A_3 X A_4A_5 X A_6A_1, les
diagonales A_1A_4, A_2A_5, A_3A_6 sont concourantes, et reciprocuement.
(L'education mathematique 49(1946-1947) 150, #8941)

Does anyone know an earlier one?

Note that the theorem was once proposed as a problem by HSMC:
Given six consecutive points A, B, C, D, E, and F on a circle, prove
that if (AB)(CD)(EF) = (BC)(DE)(FA), then AD, BE, and CF are concurrent.
(The Mathematics Student Journal, v. 27, #5, 1980, p. 3, # 527,
by H. S. M. Coxeter)

Antreas
• Is it possible that there is any connexion with Martin Gardner, The Asymmetric Propeller, Coll. Math. J., 30(1999) 18-22, and the Bankoff, Erd os, Klamkin Math
Message 2 of 3 , Jul 3, 2000
Is it possible that there is any connexion with Martin Gardner, The
Asymmetric Propeller, Coll. Math. J., 30(1999) 18-22, and the Bankoff,
Erd"os, Klamkin Math Mag article (1977?) referred to therein? R.

On Sat, 1 Jul 2000 xpolakis@... wrote:

> In the Greek mathematical periodical Diastasis [=dimension] #3-4, 1998,
> pp. 24-37, I read the article by Nikos Kyriazis:
> A New Geometry Theorem and its more Significant Applications.
>
> The author uses the theorem as a lemma to prove well-known theorems:
> The three angle-bisectors/medians/altitudes of a triangle concur, etc.
>
> Actually the theorem is not new. The earliest reference I found is the
> following:
>
> Soit A_1A_2A_3A_4A_5A_6 un hexagone convexe inscrit dans un cercle.
> Si le produit A_1A_2 X A_3A_4 X A_5A_6 des cotes de rang pair egale
> le produit des cotes de rang impair A_2A_3 X A_4A_5 X A_6A_1, les
> diagonales A_1A_4, A_2A_5, A_3A_6 sont concourantes, et reciprocuement.
> (L'education mathematique 49(1946-1947) 150, #8941)
>
> Does anyone know an earlier one?
>
> Note that the theorem was once proposed as a problem by HSMC:
> Given six consecutive points A, B, C, D, E, and F on a circle, prove
> that if (AB)(CD)(EF) = (BC)(DE)(FA), then AD, BE, and CF are concurrent.
> (The Mathematics Student Journal, v. 27, #5, 1980, p. 3, # 527,
> by H. S. M. Coxeter)
• ... I don t know of an earlier reference but I am glad you raise the point. I do, however, have two references different from yours. I will write about this
Message 3 of 3 , Jul 3, 2000
On Sat, 1 Jul 2000 xpolakis@... wrote:

> In the Greek mathematical periodical Diastasis [=dimension] #3-4, 1998,
> pp. 24-37, I read the article by Nikos Kyriazis:
> A New Geometry Theorem and its more Significant Applications.
>
> The author uses the theorem as a lemma to prove well-known theorems:
> The three angle-bisectors/medians/altitudes of a triangle concur, etc.
>
> Actually the theorem is not new. The earliest reference I found is the
> following:
>
> Soit A_1A_2A_3A_4A_5A_6 un hexagone convexe inscrit dans un cercle.
> Si le produit A_1A_2 X A_3A_4 X A_5A_6 des cotes de rang pair egale
> le produit des cotes de rang impair A_2A_3 X A_4A_5 X A_6A_1, les
> diagonales A_1A_4, A_2A_5, A_3A_6 sont concourantes, et reciprocuement.
> (L'education mathematique 49(1946-1947) 150, #8941)
>
> Does anyone know an earlier one?
>
> Note that the theorem was once proposed as a problem by HSMC:
> Given six consecutive points A, B, C, D, E, and F on a circle, prove
> that if (AB)(CD)(EF) = (BC)(DE)(FA), then AD, BE, and CF are concurrent.
> (The Mathematics Student Journal, v. 27, #5, 1980, p. 3, # 527,
> by H. S. M. Coxeter)
>
>

I don't know of an earlier reference but I am glad you raise the
point. I do, however, have two references different from yours.
now I just remember that I once saw the theorem as an excersize
for traing students for a mathematical competition.
I have seen the article of Kyriazis you quote, and I knew the
result years now.
Unfortunately Kyriazis presents it as if he discovered THE theorem
in geometry. His proof is too elaborate but here is a trivial one:
By the trigonometric form of Ceva's theorem applied to triangle
ACE we have