Geometry problem

[ins-]

It is given a quadrilateral with intersection point of the diagonals - P that is inscribed in circle with center O. M and N are the feets of the perpendiculars from P to AD and BC respectively. K and L are the intersection points of PM with BC and PN with AD respectively. Q is the intersection point of KL and OP. Prove that Q is the middle of KL.

[Kallikanzarid]

(Скрыто модератором zhoraster)

It rubs the lotion on its skin. It does this whenever it's told. It puts the lotion on its skin, or else it gets the hose again.

zhoraster:

Замечание за оффтоп!

[sergey1]

Оказывается, центры описанных окружностей треугольников $APD$ и $BCP$ лежат на прямых $NP$ и $MP$ соответственно. Поэтому $OEPF$ - параллелограмм и остается показать, что прямые $FE$ и $KL$ параллельны.

Откуда задача?

[ins-]

Thank you for the valuable idea and the time spent. A friend of mine said that is not a new problem but not said the exact source where he saw the problem. I discovered it on my own as a fact using software. I like discovering new statements and finding a solution for them. I hope you like the problem.

(Оффтоп)

/I'm sorry for my bad English but my Russian is worst./