1. The problem involves understanding and applying theorems about tangent lines, secant segments, and their lengths with respect to circles, based on given points and segments in Figures 1, 2, and 3. 2. For Figure 1, if two secant segments extend from an external point C to circle A, then according to the theorem: $$\text{(Secant 1 length)} \times \text{(External segment 1 length)} = \text{(Secant 2 length)} \times \text{(External segment 2 length)}$$ If the segments are $CB$ and $CE$ with external parts $CC$ to circle points, the product rule holds. 3. For Figure 2, involving a tangent segment and a secant segment from external point C to circle A, the theorem states: $$\text{(Tangent segment length)}^2 = \text{(Secant segment length)} \times \text{(External secant segment length)}$$ If the tangent touches at E and secant intersects at E and A, then: $$CE^2 = CA \times CE_{ext}$$ 4. For Figure 3, when two secant lines intersect inside a circle at point E, the theorem states: $$\text{(Segment 1 part 1)} \times \text{(Segment 1 part 2)} = \text{(Segment 2 part 1)} \times \text{(Segment 2 part 2)}$$ With lines CG and AR intersecting at E inside the circle: $$CE \times EG = AE \times ER$$ Steps summary: - Identify the segments (whole and external parts) based on the points. - Apply the appropriate theorem (secant-secant, tangent-secant, or intersecting secants). - Set up the equation following the product rule. - Solve for the unknown segment length if given values. Final answers are expressions relating lengths of segments as per the respective theorems: Figure 1: $$CB \times CC_{ext} = CE \times CE_{ext}$$ Figure 2: $$CE^2 = CA \times CE_{ext}$$ Figure 3: $$CE \times EG = AE \times ER$$