If your “yellow star point” is as indicated in your diagram, there is a simple formula for finding it which doesn’t require finding J and K. Define vectors B = [x1;y1], C = [x2;y2], and define rB and rC as the respective radii of the circles with centers at B and C.
BC = B-C;
S = (B+C)/2 - (rB^2-rC^2)/(2*dot(BC,BC))*BC; % <-- yellow star point
Of course, if rB and rC are equal, then S is simply
S = (B+C)/2
which would be the midpoint between B and C.
