Question

Две окружности касаются внутренним образом в точке M . Секущая m пересекает окружности последовательно в точках A , B , C , D . Докажите, что угол AMB = углу CMD.

Answer 1

Ответ:

...........................................................

Объяснение:

Answer 2

Given Information:

We are given that two circles touch internally at point M. A secant m intersects the circles sequentially at points A, B, C, and D. We need to prove that angle AMB is equal to angle CMD.

Approach:

To prove that angle AMB is equal to angle CMD, we can use the properties of tangents and secants intersecting circles. We will break down the proof into smaller steps.

Proof:

1. Let's draw the diagram with the given information. We have two circles touching internally at point M. The secant m intersects the circles at points A, B, C, and D. The diagram should look like this:

           A-----B
          /       \
         /         \
        M           \
       /             \
      /               \
     C-----------------D
   

2. Since the two circles touch internally at point M, we can draw tangents to the circles at point M. Let's label the points where the tangents intersect the secant as E and F. The diagram now looks like this:

           A-----B
          /       \
         /         \
        M           \
       / \           \
      /   \           \
     C-----E-----------D
   

3. Since the tangents are drawn from the same point M, we know that ME = MF. This is because tangents drawn from an external point to a circle are equal in length.

4. Now, let's consider the triangles AME and DME. We have: - Angle AME = Angle DME = 90 degrees (tangent is perpendicular to the radius at the point of contact) - ME = ME (common side) - ME = ME (proved in step 3)

Therefore, by the Side-Angle-Side (SAS) congruence criterion, triangles AME and DME are congruent.

5. Since triangles AME and DME are congruent, their corresponding angles are equal. Therefore, angle AME = angle DME.

6. Now, let's consider the triangles AMB and CMD. We have: - Angle AMB = Angle AME (corresponding angles of congruent triangles) - Angle CMD = Angle DME (corresponding angles of congruent triangles)

Therefore, angle AMB = angle CMD.

7. Hence, we have proved that angle AMB is equal to angle CMD.

Conclusion:

We have successfully proved that angle AMB is equal to angle CMD using the properties of tangents and secants intersecting circles.