Let and be two circles with centers and and equal radius such that . Let and be two points lying on the circle and being symmetric to each other with respect to the line . Let be an arbitrary point on . Prove that
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Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that
$$PA^2 + PB^2 \geq 2r^2.$$
Source: Međunarodna matematička olimpijada, shortlist 1967
Školjka 
and 
be two circles with centers 
and 
and equal radius 
such that 
. Let 
and 
be two points lying on the circle 
. Let 
be an arbitrary point on 