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## Homework Statement

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?