1. ABC is an equilateral triangle inscribed in a circle. Point P is any point on the arc BC (excluding B and C). Prove that: PA = PB + PC.

2. The equilateral triangle ABC is inscribed in a circle. Point P is any point on the arc BC (not containing point A). Prove that PA = PB + PC.

3. "We have drawn a triangle PQR in which ∠Q is a right angle. If S is any point on side QR, then prove that: PS² + QR² = PR² + QS²"

4. ABC is an isosceles triangle with ∠C as the right angle. If D is any point on side AB, then prove that: AD² + DB² = 2CD²

5. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?

6. Given: In a circle with center O, MN is any chord that is not a diameter, and AC is a diameter. To Prove: AC > MN, i.e., the diameter is the longest chord of a circle. Construction: From the center O, draw a perpendicular OD to the chord MN. Join O and M. Proof: OM > MD [∵ Triangle OMD is a right-angled triangle and OM is the hypotenuse] Or, OA > MD [∵ OA = OM, both are radii of the same circle] Or, ½AC > ½MN Or, AC > MN Therefore, the diameter is the longest chord of a circle.

7. Prove that if a perpendicular is drawn from the right-angled vertex of any right-angled triangle to the hypotenuse, then the two resulting triangles on either side of the perpendicular are similar to each other and each is also similar to the original triangle.

8. In triangle ABC, point O is located inside the triangle such that OA = OB and \(\angle\)AOB = 2\(\angle\)ACB. If a circle is drawn with center O and radius equal to OA, then point C will lie on the circle.

9. "Chords PQ and RS of a circle intersect each other at point X inside the circle. By joining P to S and R to Q, prove that triangles ∆PXS and ∆RSQ are similar. From this, prove that: PX × XQ = RX × XS Or, when two chords of a circle intersect internally, the product of the two segments of one chord is equal to the product of the two segments of the other chord.

10. Draw an equilateral triangle ABC with each side measuring 5 cm. Then, draw the circumcircle of that triangle. At point A on the circle, draw a tangent. On the tangent, take a point P such that AP = 5 cm. From point P, draw another tangent to the circle, and write down which point on the circle this second tangent touches.

11. From an external point A of a circle, two tangents AB and AC are drawn, touching the circle at points B and C respectively. A tangent is drawn at point X, which lies on the arc BC, and it intersects AB and AC at points D and E respectively. Prove that the perimeter of triangle ∆ADE = 2 × AB.