1. The problem statement, all variables and given/known data Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points D and Ε with MA and MB respectively. Prove that (1) the perimeter of ▲DME, and (2) the ∠DOE (where О is the center of the circle) do not depend on the position of the point C. 2. The attempt at a solution In my first attempt, I imagined the point C sliding on the smaller arc AB like a pendulum and when C is at A or B, the ∠DOE will be half of the ∠AOB. Following the same imagination, because the tangent at C will swipe out equal areas in the ▲s AOM and BOM, the perimeter of the ▲DME will remain constant. In my second attempt, I followed a hint and proved that ∠DOE is half of ∠AOB and the perimeter of ▲DME is equal to MA+MB. However, I've proved this only when the tangent at C is perpendicular to OM. Even if this proof is all that is required, how shall I prove that the perimeter of ▲DME and the ∠DOE are independent of the position of point C?