Complete quadrilateral properties

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Let $ABCD$ is a convex quadrilateral. $E$ is the intersection point of the lines $AB$ and $CD$. $F$ is the intersection point of the lines $AD$ and $BC.$ $P$ is the intersection point of the diagonals $AC$ and $BD$. Through $P$ is drawn a line intersecting the line $EF$ at the point $P'$. Through the vertices $A$, $B$, $C$, $D$ are drawn lines parallel to the line $PP'$ intersecting the line $EF$ at the points $A'$, $B'$, $C'$, $D'$, respectively. Prove that:
a) $\frac{1}{AA'}+\frac{1}{BB'}+\frac{1}{CC'}+\frac{1}{DD'} = \frac{4}{PP'}$;
b) $\frac{1}{AA'}+\frac{1}{CC'}=\frac{1}{BB'}+\frac{1}{DD'}$.

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It is also true that: $\frac{1}{AA'}+\frac{1}{CC'} = \frac{1}{BB'}+\frac{1}{DD'}$.

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If a moderator read this - can he/she change the title of this topic to "Complete quadrilateral properties" and to rewrite the dependencies as a) and b) conditions?

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Hint:

(Оффтоп)

http://www.qbyte.org/puzzles/p002s.html i think by using this problem and some interesting constructions and calculations we can solve our problem.

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http://www.math10.com/f/viewtopic.php?f=49&t=11926 - you can see another approach. No additional constructions needed - just similar triangles and Menelaus. There are at least 4 ways to solve the problem. I hope you like it.