1. I have drawn a triangle ABC in which AB = AC, and E is any point on the extended line BC. If the circumcircle of ∆ABC intersects AE at point D, prove that ∠ACD = ∠AEC.

2. In the center circle O, AB is a diameter. C is any point on the circumference of the circle where AC = 3 cm and BC = 4 cm. Find the length of AB.

(a) 3 cm (b) 4 cm (c) 5 cm (d) 7 cm

3. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

4. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

5. If in triangle \( \triangle ABC \), the sum of the medians AD, BE, and CF is \(x\), and the sum of the sides is \(y\), then the relationship between \(x\) and \(y\) is —

(a) \(x\gt y\) (b) \(x\lt y\) (c) \(x= y\) (d) None of the above

6. 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?

7. 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.

8. AB is the diameter of a circle centered at O. C is any point on the circle. Given ∠BAC = 50° and CD is perpendicular to AB, Find the value of ∠BCD.

9. 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.

10. In \( \triangle ABC \), points \( D \) and \( E \) lie on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \) and \( AD:DB = 3:1 \). If \( EA = 33 \) cm, then the length of \( AC \) is _____.

11. In a circle centered at O, AB is a diameter, and C is any point on the circle. Given that \(\angle\)OAC = 45°, determine the measure of \(\angle\)OCB.

(a) 90° (b) 45° (c) 30° (d) 60°

12. Draw a median \(AD\) of \(\triangle ABC\). If a straight line parallel to \(BC\) intersects \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that \(AD\) bisects the line segment \(PQ\).

13. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters.
The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\).
Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters.

From \( \triangle \)ABO, we get:
\(\cfrac{AB}{BO} = \tan \theta\)
Or, \(\cfrac{x}{60} = \tan\theta\) --------(i)

From \( \triangle \)COD, we get:
\(\cfrac{CD}{OD} = \tan(90^o - \theta)\)
Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii)

Multiplying equations (i) and (ii), we get:
\(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\)
Or, \(\cfrac{2x^2}{60 \times 60} = 1\)
Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\)
Or, \(x = 30\sqrt2\)

\(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters.
And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters.
(Proved).

14. 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.

15. Draw an equilateral triangle with side length 7 cm. Then draw both the circumcircle and the incircle of the triangle. Using a scale, determine the lengths of the circumradius and the inradius. Also, write whether there is any relationship between them.

16. “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.”

17. In the adjacent figure, AB is the diameter of a circle centered at O. Point C is any point on the circle. Given that ∠BAC = 50° and CD is perpendicular to AB, find the value of ∠BCD.

18. We have drawn a median \(AD\) of triangle \( \triangle ABC \). If a straight line parallel to \(BC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that the line segment \(PQ\) is bisected by the median \(AD\).

19. 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.

20. AB is a diameter of a circle with center O. P is any point on the circle. Tangents are drawn at points A and B, which are intersected by the tangent drawn from point P at points Q and R respectively. If the radius of the circle is \(r\), prove that \(PQ \cdot PR = r^2\).

21. AB is the diameter of a circle with center O. A straight line drawn from point A intersects the circle at point C, and the tangent drawn from point B intersects this line at point D. Prove that: For any such straight line, the area of the rectangle formed by AC and AD is always constant.

22. "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.

23. Points \(X\) and \(Y\) are taken on sides \(PQ\) and \(PR\) of triangle \(∆PQR\), respectively. Given: \(PQ = 8\) units, \(YR = 12\) units, \(PY = 4\) units, and the length of \(PY\) is 2 units less than the length of \(XQ\). Determine, with reasoning, whether segment \(XY\) is parallel to side \(QR\).

24. 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.

25. 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.

26. On a circle with center O, A is a point on the circle. A tangent is drawn at point A, and point X is any point on that tangent. A secant drawn from point X intersects the circle at points Y and Z. If P is the midpoint of segment YZ, prove that XAPO (or XAOP) is a cyclic quadrilateral.

27. ABCD is a cyclic quadrilateral. If the sides AB and DC are extended, they meet at point P; and if the sides AD and BC are extended, they meet at point R. The circumcircles of triangles BCP and CDR intersect at point T. Prove that the points P, T, and R lie on a straight line.

28. Here’s the English translation: Inside a circle with center O, point P is any point. The radius of the circle is 13 cm, and OP = 12 cm. Find the minimum length of a chord that passes through point P.

(a) 5 cm (b) 6 cm (c) 6.5 cm (d) 10 cm