Proof:

To prove that the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact, we can use the properties of tangents and chords in circles.

Given:
Two concentric circles, with the smaller circle inside the larger circle.

To prove:
The chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

Proof:

Step 1: Draw the diagram

Draw two concentric circles, with the smaller circle inside the larger circle. Label the center of the circles as O, the point of contact between the chord and the smaller circle as A, and the midpoint of the chord as M.

Step 2: Establish the relationship between the chord and the tangent

Since the chord of the larger circle touches the smaller circle, it is tangent to the smaller circle at point A. This means that the line segment OA is perpendicular to the chord.

Step 3: Use the properties of tangents and chords

In a circle, a tangent is perpendicular to the radius drawn to the point of tangency. Therefore, the line segment OA is perpendicular to the chord. This implies that triangles OAM and OBM are right-angled triangles.

Step 4: Prove that the chord is bisected at the point of contact

Since OAM and OBM are right-angled triangles, we can use the property of right-angled triangles that the hypotenuse is twice the length of the altitude drawn to it from the right angle. In this case, the hypotenuse is the chord and the altitude is the line segment OA.

Therefore, we have:
OA = AM = MB

This shows that the chord is bisected at the point of contact, as the line segment AM is equal in length to the line segment MB.

Conclusion:

The chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. This is proved using the properties of tangents and chords in circles.