BCho nửa đường tròn (O;R) đường kính AB. Trên cùng một nửa mặt phẳng bờ AB, dựng các tiếp tuyến Ax, By của nửa đường tròn. Lấy một điểm M trên nửa đường tròn (O). Tiếp tuyến tại M của (O) cắt Ax, By lần lượt tại D, C tia AM, BM kéo dài cắt By, Ax lần lượt tại F, E. Thực hiện các yêu cầu sau:

To solve this geometry problem involving a semicircle and its tangent lines, let's follow the outlined requirements and steps.

1. **Chứng minh (Proof)**

a) To show that points D, M, O and C, M, O lie on the same circle, we can prove that they are concyclic. This can be done by showing that the angles subtended by these points are equal.

b) For point D to be the foot of the perpendicular from M to line AE, we can use the properties of tangents and radii to demonstrate that angle ADM is a right angle.

c) To prove that triangle ACB is similar to triangle ABDE, we can use the angle-angle similarity criterion (AA similarity). Show that ∠CAB = ∠ABD and ∠BCA = ∠DBA.

d) For triangle ABC and ABD, applying the property \( R^2 = AB \cdot BC = CD \), we look at the relationship between the sides and angles to show that the triangles are similar or proportional.

2. **Dựng M H ⊥ AB**

To conclude that M H is perpendicular to AB, we can use the fact that the tangent at any point of a circle is perpendicular to the radius at that point.

These steps will help you derive the necessary results both geometrically and algebraically. Ensure to provide detailed reasoning for each geometric claim, using established theorems and axioms where applicable.