1. **Problem statement:** Show that triangles ABC and FEC are similar and explain why line OB is parallel to line PE. 2. **Similarity of triangles ABC and FEC:** - Triangles are similar if their corresponding angles are equal. - Since A, B, C, D lie on the big circle with center O, and C, E, F lie on the smaller circle with center P, and AOCPF and BCE are straight lines, we analyze angles. 3. **Step 1: Identify equal angles** - Angle ABC and angle FEC subtend the same arc BC in their respective circles. - Therefore, \(\angle ABC = \angle FEC\). 4. **Step 2: Identify another pair of equal angles** - Since AOCPF is a straight line, \(\angle BAC = \angle CFA\) because they are alternate angles formed by the intersecting chords. 5. **Step 3: Third angle equality** - By the angle sum property of triangles, the third angles are equal. 6. **Conclusion:** - Triangles ABC and FEC have all corresponding angles equal, so they are similar by the AA (Angle-Angle) criterion. 7. **Why line OB is parallel to line PE:** - O and P are centers of the big and small circles respectively. - OB is a radius of the big circle, and PE is a radius of the small circle. - Since the circles touch externally at C, the line joining their centers O and P passes through C. - Lines OB and PE are both perpendicular to the tangent at point C. - Therefore, OB is parallel to PE because they are both perpendicular to the same line (the tangent at C). **Final answers:** (a) Triangles ABC and FEC are similar by AA similarity because corresponding angles subtend the same arcs and alternate angles are equal. (b) Lines OB and PE are parallel because they are both perpendicular to the tangent line at point C.