Exotic Circle

[ins-]

It is given circle k(O,R). From a point A is drawn the tangent to k. On the tangent is chosen point B (AB<R). From B is drawn a perpendicular that intersects k at the points C and D (C is between B and D). OC intersects AD at the point P. Q is the intersection point of the perpendicual from O to AD and the segment CD. Prove that OP=PQ.

[gris]

Углы, равнобедренные и прямоугольные треугольники.

Да, это ключевые слова Задача красивая. Используется много свойств, как и в предыдущей. Центральные, вписанные улы, углы между касательной и хордой. Доступна девятитиклассникам по стандарту. Но до олимпиадной, конечно, не дотягивает.

$\angle ODP=90^{\circ}-\frac12\angle DOA=90^{\circ}-\angle DAB=\angle DAB$. Дальше понятно.

Кстати, Точка $C$ лишняя. Можно выбрать произвольную $P$ на $AD$.

[ins-]

Можно послать Ваши решения обе проблема? Или это ключевые слова?

[ins-]

gris ... you are right ... I forced myself to invent some problems on circumferences - one of my weakest places when I invent geometry problems ... I'm tired maybe it is the reason. I'll stop with inventing for a while. When I have inspiration I'll show more problems.

[gris]

Inspiration prays: Please, don't stop!
What the reason is to stop if you have invented a good problem but of training level? It will be useful anyway. If you are building a whole bunch of circle-tangent problems there are of course pieces of different levels of the difficulty and of the grace, but it is usually normal to make masterpieces through a long long way to go of quite dull exercises.
What I'm talking about? Oh, yes, indeed. Try(1001).

[ins-]

gris, thank you for the good words. You are right. I should be more patient. I'll continue to invent new problems but not "as goal" I mean - "today I'll invent 2 problems related to circle and tangents". Math requires hard work, creativity, talent and sometimes ... inspiration and good luck.