1. **Problem statement:** Solve the problem of finding the lengths of segments formed by two intersecting chords inside a circle without using the theorem "When two chords intersect inside a circle, the products of the segments of each chord are equal." 2. **Understanding the problem:** Normally, the theorem states that if two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. Since we cannot use this theorem, we need to find another approach. 3. **Alternative approach:** Use coordinate geometry or properties of similar triangles formed by the chords and the circle. 4. **Step-by-step solution:** - Place the circle on a coordinate plane with a known center and radius. - Assign coordinates to the points where the chords intersect the circle. - Use the distance formula to express the lengths of the chord segments. - Use the fact that the chords intersect at a point inside the circle, and set up equations based on the coordinates. - Solve the system of equations to find the lengths of the segments. 5. **Summary:** By using coordinate geometry and distance formulas, we can find the lengths of the chord segments without relying on the chord segment product theorem. **Note:** Without specific numerical values or a diagram, this is the general method to solve such a problem without the theorem.