Get the pixels on the boundary with bwboundaries or bwtraceboundary. Calculate their distance to the centre of your circle. Then the absolute difference of that distance with the radius of that circle. Anything that is smaller than an arbitrary small value is your intersection points. Something like:
%input variables:
% yourimage: the binary image
% row_wheelcentre and col_wheelcentre: coordinates of centre of wheel/circle
% radius: the radius of the circle
%output variables
% intersect point: Nx2 matrix containing the N points of the wheel intersecting the circle
boundaries = bwboundaries(yourimage, 'nohole');
%assuming that only one object is detected, or that the gear is the first one:
gearboundary = boundaries{1};
distances = hypot(gearboundary(:, 1) - row_wheelcentre, gearboundary(:, 2) - col_wheelcentre);
intersectpoint = gearboundary(abs(distances - radius) < 1e-5, :); %arbitrary 1e-5 threshold may need adjusting
