1. **Problem Statement:** Given a parallelogram ABFD with points D, A, B, and C on a circle, and lines BF and DF extended to meet DC and CB at points E and G respectively, with DC perpendicular to EB. We need to: - 9.1 Show that AC is a diameter of the circle. - 9.2 Prove that BGED is a cyclic quadrilateral. --- 2. **Key Formulas and Rules:** - A diameter subtends a right angle to any point on the circle (Thales' theorem). - Opposite angles of a cyclic quadrilateral sum to 180°. - In a parallelogram, opposite sides are parallel and equal. - If two lines are perpendicular, their angles are 90°. --- 3. **Step 9.1: Show AC is a diameter** - Since D, A, B, C lie on the circle, quadrilateral DABC is cyclic. - ABFD is a parallelogram, so AB is parallel and equal to DF. - Because DC ⊥ EB and E lies on the extension of BF, angle DEB = 90°. - By Thales' theorem, if angle DAB subtended by AC is 90°, then AC is a diameter. To prove angle DAB = 90°: - Since ABFD is a parallelogram, AB is parallel to DF. - Angles DAB and DFB are alternate interior angles, so angle DAB = angle DFB. - Angle DFB is subtended by chord DB in the circle. - Since D, A, B, C lie on the circle, angle DAB + angle DCB = 180° (opposite angles in cyclic quadrilateral). - But DC ⊥ EB implies angle DCB = 90°, so angle DAB = 90°. Therefore, AC subtends a right angle at D and A, so AC is a diameter. --- 4. **Step 9.2: Prove BGED is cyclic** - To prove BGED is cyclic, show that opposite angles sum to 180° or that all points lie on a circle. - Given DC ⊥ EB, angle DEB = 90°. - Since E lies on DC and BF extended, and G lies on CB extended, consider triangles and angles formed. - In parallelogram ABFD, BF is parallel to AD. - Using angle chasing: - Angle BGE = angle BFD (corresponding angles since DF extended meets CB at G). - Angle BFD = angle BAD (alternate interior angles in parallelogram). - Angle BAD + angle BED = 180° (since D, A, B, C cyclic and E lies on extension). - Therefore, angle BGE + angle BED = 180°, so BGED is cyclic. --- **Final answers:** - 9.1 AC is a diameter of the circle because it subtends a right angle at point D. - 9.2 BGED is a cyclic quadrilateral because opposite angles sum to 180°.