Introduction: A key circle geometry topic. You must memorize the two main theorems and their converses. Exam questions ("riders") will often require you to combine these theorems with other circle properties (like angles in the same segment) or properties of parallel lines.

The "S Pass" Foundation (නියත S එකකට)

• Prompt 1 (Define): What is a cyclic quadrilateral?

• Prompt 2 (State): State the theorem about the opposite angles of a cyclic quadrilateral.

• Prompt 3 (Apply): In cyclic quadrilateral PQRS, if P^=70∘, find the size of R^.

• Prompt 4 (State): State the theorem about the exterior angle of a cyclic quadrilateral.

• Prompt 5 (Apply): The side AB of a cyclic quadrilateral ABCD is produced to E. If A^=100∘, what is the size of the exterior angle CB^E?

Climbing to a "C" (C එකට පාර)

• Prompt 1 (Problem Solve): In cyclic quadrilateral ABCD, A^=2x, B^=3x, and C^=7x. Find the value of x and the size of D^.

• Prompt 2 (Proof): AB is a diameter of a circle. C and D are points on the circumference. Prove that ABCD is a cyclic quadrilateral.

• Prompt 3 (Problem Solve): In the figure, ABCD is a cyclic quadrilateral. AD is parallel to BC. If AB^C=80∘, prove that triangle ADC is isosceles.

• Prompt 4 (Proof): Prove that an isosceles trapezium is a cyclic quadrilateral.

• Prompt 5 (Application): Two circles intersect at P and Q. A straight line APB meets the circles at A and B. A straight line CQD meets the circles at C and D. Prove that AC is parallel to BD.

Aiming for a "B" (B ඉලක්කය)

• Prompt 1 (Rider): Prove that the angle bisectors of the angles of a cyclic quadrilateral form another cyclic quadrilateral.

• Prompt 2 (Rider): A circle is drawn through the vertices of a triangle ABC. The angle bisector of angle A meets the circle at P. Prove that PB = PC.

• Prompt 3 (Rider): ABCD is a parallelogram. The circle passing through A and B intersects AD and BC at P and Q respectively. Prove that PQCD is a cyclic quadrilateral.

• Prompt 4 (Synthesis): In cyclic quadrilateral ABCD, the side AB is produced to E. The bisector of angle CBE meets the bisector of angle ADC at P. Prove that AD^P+AB^P=180∘.

• Prompt 5 (Formal Proof): Provide a formal proof for the theorem: "The opposite angles of a cyclic quadrilateral are supplementary."

Securing the "A" Distinction (A සාමාර්ථය තහවුරු කරගන්න)

• Prompt 1 (Challenge Rider): ABCD is a cyclic quadrilateral. The angle bisectors of angles A, B, C, D meet the circle at P, Q, R, S respectively. Prove that the quadrilateral PQRS has its diagonals perpendicular to each other.

• Prompt 2 (Challenge Rider): Two circles intersect at points X and Y. A straight line is drawn through X to meet the circles at A and B. The tangents at A and B intersect at a point C. Prove that the quadrilateral AYBC is cyclic.

• Prompt 3 (Ptolemy's Theorem): For a cyclic quadrilateral ABCD, prove that the sum of the products of the opposite sides is equal to the product of the diagonals (i.e., AB⋅CD+BC⋅DA=AC⋅BD). This is a very advanced rider.

• Prompt 4 (Synthesis): A triangle PQR is inscribed in a circle. The altitude from P meets the circle at S, and the altitude from Q meets the circle at T. The orthocenter is H. Prove that R is the midpoint of the arc ST.

• Prompt 5 (Problem Solve): In cyclic quadrilateral ABCD, AB=BC, CD=DA. Prove that AC is a diameter of the circle.