Take a fixed point A in the plane. Draw all circles with center A. Consider any circle. Draw tangents at each point of the circle. Now pair up the tangents perpendicular to each other. Do this for all the circles. Also pair up the perpendicular lines through A.

Set up an x-y coordinate system. It suffices to prove that it is possible to pair up the lines parallel to the x- and y- axes. This can be done by pairing up the horizontal and vertical lines through the point $(a,a)$.