Introduction: This lesson introduces three more crucial circle theorems. The 'Alternate Segment Theorem' is often found challenging but is very powerful. Be prepared for riders that combine tangent properties with those of cyclic quadrilaterals or similar triangles.

The "S Pass" Foundation (නියත S එකකට)

• Prompt 1 (State): State the theorem about the angle between a tangent and a radius at the point of contact.

• Prompt 2 (State): State the theorem about the two tangents drawn to a circle from an external point.

• Prompt 3 (Apply): Tangents PA and PB are drawn from a point P to a circle. If PA = 15 cm, what is the length of PB?

• Prompt 4 (State): State the Alternate Segment Theorem.

• Prompt 5 (Apply): A tangent PQ touches a circle at A. Chord AB is drawn such that angle QA^B=70∘. If C is a point in the major segment, what is the size of angle AC^B?

Climbing to a "C" (C එකට පාර)

• Prompt 1 (Problem Solve): Two tangents PA and PB are drawn from a point P to a circle with centre O. If angle AP^B=80∘, find the size of angle AO^B.

• Prompt 2 (Proof): Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

• Prompt 3 (Problem Solve): The sides AB, BC, and CA of a triangle ABC touch a circle at P, Q, and R respectively. If AB=10, BC=8, and CA=12, find the lengths of AP, BQ, and CR.

• Prompt 4 (Proof): In triangle ABC, the angle bisector of angle A meets the circumcircle at P. Prove that the tangent at P is parallel to BC.

• Prompt 5 (Application): Two circles touch externally at a point P. A straight line is drawn through P to meet the circles at A and B. Prove that the tangents at A and B are parallel.

Aiming for a "B" (B ඉලක්කය)

• Prompt 1 (Rider): A circle is inscribed in a quadrilateral ABCD, touching the sides at P, Q, R, S. Prove that AB+CD=AD+BC.

• Prompt 2 (Rider): Two circles touch internally at a point T. A chord AB of the larger circle touches the smaller circle at P. Prove that TP bisects the angle AT^B.

• Prompt 3 (Rider): A common tangent AB is drawn to two circles that touch each other externally at C. Prove that angle AC^B=90∘.

• Prompt 4 (Synthesis): From an external point P, tangents PA and PB are drawn to a circle. The line PC passes through the centre O and intersects the circle at C. Prove that AC bisects the angle PA^B.

• Prompt 5 (Formal Proof): Provide a formal proof for the theorem: "The angles which a tangent to a circle makes with a chord drawn from the point of contact are respectively equal to the angles in the alternate segments of the circle".

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

• Prompt 1 (Challenge Rider): Two circles intersect at A and B. A common tangent touches the circles at P and Q. Prove that the line AB bisects PQ.

• Prompt 2 (Challenge Rider): In a right-angled triangle ABC, a circle is drawn with the hypotenuse AC as diameter. The tangent at B intersects AC produced at P. Prove that triangle PBC is an isosceles triangle.

• Prompt 3 (Direct Common Tangent): Two circles with centres O1 and O2 and radii r1 and r2 touch each other externally. Derive the formula for the length of the direct common tangent.

• Prompt 4 (Synthesis): Let ABC be an acute-angled triangle. Let H be its orthocentre. Let L, M, N be the feet of the altitudes from A, B, C respectively. Prove that the circumcircle of triangle LMN is the nine-point circle of triangle ABC.

• Prompt 5 (Problem Solve): In a cyclic quadrilateral ABCD, the tangent at A is parallel to the diagonal BD. Prove that AC bisects the angle BC^D.