Two undirected red line segments are placed diametrically opposite along diameter on a unit circle of radius OF such that the product
$$ Uu\cdot Vv= OF^2 $$ remains a constant thus mapping U to V.
If the position of U is given then what geometric construction determines the position of V?
Thanks in advance.
EDIT1:
The following is one way of construction. OC= 2 OF for two concentric circles.
Reflect U to point M on the other side and draw circle through (C,O,M), (CM is diameter) which is cut by a horizontal line through F at N. The circle drawn through (N, F, u) cuts the x-axis at the required point P.
However still looking for a single step or simpler procedure for a single Circle between U and V.