Final answer:
The question involves proving a geometric relationship within a circle but lacks sufficient details for a specific proof. However, the Power of a Point Theorem is typically used to prove such products of segment lengths.
Step-by-step explanation:
The provided mathematical geometry problem involves a circle, various line segments, and angles, aiming to prove a relationship between the lengths of these segments. Unfortunately, the references given are disjointed and do not seem to pertain directly to the query about circle k(O). However, based on standard geometric theorems and properties related to circles and triangles, one could infer that the Power of a Point Theorem might be used in proving such a relationship, where the product of the lengths of two segments from a point outside a circle to the points of tangency with the circle is constant. This theorem is reflected in the equation AD·CB = AC·CD, suggesting that points A, B, C, and D lie on the circle or its tangents.
In a typical geometry problem like this, additional details such as the positions of points A, B, C, and D about circle k(O) and each other would be necessary to provide a step-by-step proof. For instance, if points A and B were on a tangent to the circle and points C and D were points of tangency, the proof would proceed by showing that the triangles formed by these points are similar, and therefore, the product of the segments in question remains constant.
When complete information is provided, the proof can leverage geometrical postulates and theorems such as similar triangles, tangent-secant theorem, or the Pythagorean Theorem—the latter represented by the formula a² + b² = c². Regrettably, without the correct information pertaining to the circle and the location of the points, providing an accurate proof is not feasible.