Dedicated to gris

[ins-]

I would like to thank to gris for the good words and encouraging me to invent new problems.
Few minutes ago I (re)discovered the statement below. I don't know if it is new or old, easy or hard,
beautiful or ugly problem, but I think it can be interesting for someone.

It is given a semicircle k with diameter AB. It is chosen a random point C on k different from A and B.
D is the middle of AC. BD intersects k at E. A line through E parallel to AC intersects k at the point F.
P is the intersecting point of AB and FD. In what ratio P divides AB?

[venco]

(Оффтоп)

У меня сложилось ощущение, что Вы программу написали, которая подобные построения ищет. В принципе, искать их несложно, трудность вызывает, на мой взгляд, отфильтровать нетривиальные.

[gris]

[ins-]

(Оффтоп)

I'm using software but it don't find such constructions. I use standart functions of existing software and my intuition and lots of experiments. It is the reason for different levels of difficulty of my problems. To find how difficult is a problem and if it is known I'm posting them to some math sites. It is very useful way to find different opinions and approaches. I think it is the place that computer knowledge and math can collaborate and some new problems can be born. I'm happy if someone like and use some of my problems. The beauty is because no one can know how difficult is a given problem and the creativity of the people is increased.

[Dimoniada]

$BP=2*PA$

[ins-]

It is the answer but why?

[ins-]

gris, in the picture you gave some hints. can you give the details required for a complete solution?

Is the problem well known?
Is it beautiful?
What about its level of difficulty?

[ins-]

There are two solutions. The second one implements gris' idea.
http://www.math10.com/f/viewtopic.php?f=10&t=3128
http://www.artofproblemsolving.com/Foru ... 8&t=366820