Suppose m is between OA, because the angle CBE >；; 90 degrees, so it can only be BC=BE, let E(m, n), from the title we know B (0 0,2) and C (0 0,4), so BC=2, so BE=2. Two equations can be listed from the distance formula of two coordinates, plus two straight lines intersect at point E, and t can be obtained by solving a ternary quadratic equation.

Suppose m is on the left side of point O, because the angle LBC is between 60 and 90 degrees, there are three situations that △CBE is an isosceles triangle, BC=BE, or BE=CE, or BC=CE. Similarly, use the method described in 1 to find the t value.

It's too much trouble to type formula symbols online, so I can only talk about my own ideas.