1. Three equal circles touch each other externally. Prove that the centers of these three circles form the vertices of an equilateral triangle.

2. Two identical circles, each with radius \(r\), intersect in such a way that each circle passes through the center of the other. The centers of the circles are labeled A and B, and they intersect at points P and Q. The area of triangle \(\triangle APB\) will be:

(a) \(\cfrac{\sqrt3}{4}r^2\) (b) \(\cfrac{\sqrt3}{2}r^2\) (c) \(\cfrac{\sqrt3}{3}r^2\) (d) \(\sqrt3 r^2\)

3. Two identical circles, each with a radius of 10 cm, intersect each other, and the length of their common chord is 12 cm. Find the distance between the centers of the two circles.

4. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

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. Draw an equilateral triangle with each side measuring 7 cm. Then, draw the incircle (inscribed circle) of that triangle.

7. Two identical circles, each with a radius of 13 cm, intersect each other, and the length of their common chord is 10 cm. What is the distance between the centers of the two circles?

8. Two equal circles, each with a radius of 10 cm, intersect, and the length of their common chord is 12 cm. Determine the distance between the centers of the circles.

9. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar, and each of these triangles is similar to the original triangle.

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

11. A group had ₹195 in funds. After each of its members contributed an amount equal to the number of members, the total amount was distributed equally among all members such that each received ₹28. Using Śrīdharāchārya’s formula (the quadratic formula), find the number of members in the group.

12. Two equal circles, each with a radius of 10 cm, intersect each other. The length of their common chord is 12 cm. Find the distance between their centers.

13. Three jars with equal diameter and height were partially filled with dilute sulfuric acid: - the first jar was filled to \(\frac{2}{3}\) of its height, - the second to \(\frac{5}{6}\), and - the third to \(\frac{7}{9}\). If the acid from these three jars is poured into a single jar with a diameter of 2.1 decimetres, and the height of the acid in this jar becomes 4.1 decimetres, then calculate and write the height of each of the original three jars, given that their diameter was 1.4 decimetres.

14. “Two equal circles are such that the center of one lies on the other, and they intersect at points A and B. A straight line through A intersects the circles again at points C and D. Prove that triangle BCD is equilateral.”

15. If a perpendicular is drawn from the right-angled vertex of any right triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other, and each of them is also similar to the original triangle.

16. 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.

17. Draw an isosceles triangle in which each of the equal sides is 7 cm and the base is 8 cm long. Then, draw an incircle of that triangle.

18. Draw an isosceles triangle whose base is 7.8 cm and the length of each of the equal sides is 6.5 cm. Then draw an incircle (an inscribed circle) inside that triangle.

19. Draw an isosceles triangle with a base of 6.5 cm and two equal sides of 7 cm each. Then, draw the incircle of that triangle.