Нерешаемая задача

[Keter]

Задача: точки $A, B, C, D$ лежат на одной окружности с центром О. Прямые $AB$ и $CD$ пересекаются в точке E. Окружности, описанные вокруг треугольников $AEC$ и $BED$ пересекаются в точках $E$ и $P$. Доказать, что точки $A, D, P, O$ принадлежат одной окружности.

Можете проверить, но это всегда так и при внешнем пересечении прямых.
Как её доказать???!!

[ins-]

B, C, P, O probably are also concyclic. It remains a Miquels' theorem http://nl.wikipedia.org/wiki/Bestand:PuntvanMiquel.PNG <EPO is right which is probably more beautiful result. Thanks for the beautiful problem. You can use these materials to prove your problem:

http://webcache.googleusercontent.com/s ... clnk&gl=bg

It is also true that EP pass through the intersection point of the diagonals quadrilateral ABCD. If we manage to prove that - your problem is solved.

You can see a similar problem: http://imomath.com/othercomp/Bul/BulMO496.pdf problem 5.

[Keter]

ins- в сообщении #478813 писал(а):

B, C, P, O probably are also concyclic. It remains a Miquels' theorem http://nl.wikipedia.org/wiki/Bestand:PuntvanMiquel.PNG <EPO is right which is probably more beautiful result. Thanks for the beautiful problem. You can use these materials to prove your problem:

http://webcache.googleusercontent.com/s ... clnk&gl=bg

It is also true that EP pass through the intersection point of the diagonals quadrilateral ABCD. If we manage to prove that - your problem is solved.

You can see a similar problem: http://imomath.com/othercomp/Bul/BulMO496.pdf problem 5.

Thank you! Bulgarian I really do not know, but anyway)

[ins-]

Bulgarian is similar to Russian. They are Slavic languages.