Cho 2 đường tròn tâm O, bán kính R và tâm O' bán kính r tiếp xúc với A và R khác r. Đường nối OO' tại B và C. Đường thẳng A lần lượt tiếp xúc với 2 đường tròn OO' tại D và E. Gọi M là giao điểm của BD và CE. Chứng minh:

To prove the statements given in the geometry problem involved with two circles, you can follow these steps:

### a) Proving \( \angle DME = 90^\circ \)

1. **Properties of Tangent Lines**: Since \( AD \) is tangent to the circle centered at \( O \) at point \( D \), the radius \( OD \) is perpendicular to the tangent line at the point of tangency, which gives us \( \angle ODB = 90^\circ \).

2. Similarly, since \( AE \) is tangent to the circle centered at \( O' \) at point \( E \), the radius \( O'E \) is perpendicular to the tangent line, which means \( \angle O'EB = 90^\circ \).

3. Since \( B \) lies on segment \( OO' \) and both \( D \) and \( E \) are points of tangency, triangles \( ODB \) and \( O'EB \) are similar in configuration. Therefore, the angles formed at point \( M \) where lines \( BD \) and \( CE \) intersect are such that \( \angle DME + \angle ODB + \angle O'EA = 90^\circ + 90^\circ = 180^\circ \).

4. Therefore, \( \angle DME = 90^\circ \).

### b) Proving \( MA \) is tangent to circle \( O' \)

1. Because \( A \) is the point of tangency for both circles, the length \( MA \) which connects points \( M \) and \( A \) is perpendicular to radius \( O'M \).

2. This means \( MA \) is tangent to the circle at the point \( E \) implying the condition for tangency holds.

### c) Proving \( MD \cdot MB = ME \cdot MC \)

1. From part (a), we established that \( \angle DME = 90^\circ \) thus triangles \( MDE \) and \( MBE \) are formed.

2. By the property of the angles from tangents forming equal products in intersecting secants, we get that \( MD \cdot MB = ME \cdot MC \).

### Conclusion

The conditions of tangency and the properties of angles and circles help to fulfill each of the requirements laid out in the problem. Each part logically follows from established geometric principles involving circles and tangents.