{
  "title": "Cyclic Quadrilateral - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Sum of opposite angles in cyclic quadrilateral is:", "options": {"A": "90°", "B": "180°", "C": "270°", "D": "360°"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "Which quadrilateral is always cyclic?", "options": {"A": "Parallelogram", "B": "Rectangle", "C": "Rhombus", "D": "Trapezium"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "In cyclic quadrilateral ABCD, if ∠A=70°, then ∠C=?", "options": {"A": "70°", "B": "90°", "C": "110°", "D": "130°"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "Exterior angle of cyclic quadrilateral equals:", "options": {"A": "Adjacent interior angle", "B": "Opposite interior angle", "C": "Sum of opposite angles", "D": "Difference of opposite angles"}, "correct_answer": "B"},
   {"id": 5, "difficulty": "easy", "question": "If all vertices of quadrilateral lie on circle, it is called:", "options": {"A": "Concyclic", "B": "Cyclic", "C": "Both A and B", "D": "Circumscribed"}, "correct_answer": "C"}, {"id": 5, "difficulty": "easy", "question": "If all vertices of quadrilateral lie on circle, it is called:", "options": {"A": "Concyclic", "B": "Cyclic", "C": "Circumscribed", "D": "B and C"}, "correct_answer": "D"},
    {"id": 6, "difficulty": "easy", "question": "In cyclic quadrilateral, angle in semicircle is:", "options": {"A": "45°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "Ptolemy's theorem relates sides and diagonals of:", "options": {"A": "Any quadrilateral", "B": "Cyclic quadrilateral", "C": "Parallelogram", "D": "Trapezium"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "Which test proves quadrilateral cyclic?", "options": {"A": "Opposite angles supplementary", "B": "Exterior angle equals interior opposite", "C": "Both A and B", "D": "None"}, "correct_answer": "C"},
    {"id": 9, "difficulty": "easy", "question": "In cyclic quadrilateral, sum of all angles is:", "options": {"A": "180°", "B": "270°", "C": "360°", "D": "540°"}, "correct_answer": "C"},
    {"id": 10, "difficulty": "easy", "question": "If one side of cyclic quadrilateral is extended, exterior angle equals:", "options": {"A": "Adjacent interior angle", "B": "Interior opposite angle", "C": "Sum of interior angles", "D": "None"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "Cyclic parallelogram must be:", "options": {"A": "Rectangle", "B": "Rhombus", "C": "Square", "D": "Any parallelogram"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "If quadrilateral has angles 70°,95°,85°,110°, is it cyclic?", "options": {"A": "Yes", "B": "No", "C": "Can't say", "D": "If modified"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "In cyclic quadrilateral, perpendicular bisectors of sides are:", "options": {"A": "Concurrent at center", "B": "Parallel", "C": "Perpendicular", "D": "Not concurrent"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Maximum number of right angles in cyclic quadrilateral:", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "If three vertices of quadrilateral lie on circle and opposite angles supplementary, fourth vertex:", "options": {"A": "Lies on same circle", "B": "Lies outside", "C": "Lies inside", "D": "Can't determine"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Cyclic trapezium is always:", "options": {"A": "Isosceles", "B": "Right angled", "C": "Scalene", "D": "Equilateral"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "In circle, quadrilateral with equal opposite angles is:", "options": {"A": "Rectangle", "B": "Parallelogram", "C": "Rhombus", "D": "Trapezium"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "If ∠A+∠C=180° in quadrilateral ABCD, points A,B,C,D are:", "options": {"A": "Concyclic", "B": "Collinear", "C": "Vertices of parallelogram", "D": "None"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Ptolemy's theorem: For cyclic quadrilateral ABCD,", "options": {"A": "AB×CD + BC×DA = AC×BD", "B": "AB×BC + CD×DA = AC×BD", "C": "AB×CD + AD×BC = AC×BD", "D": "AB×AD + BC×CD = AC×BD"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "Which is not necessarily cyclic?", "options": {"A": "Square", "B": "Rectangle", "C": "Isosceles trapezium", "D": "Rhombus"}, "correct_answer": "D"},
   {"id": 21, "difficulty": "medium", "question": "In cyclic quadrilateral ABCD, ∠A=2x, ∠B=3x, ∠C=4x, ∠D=5x. Find x.", "options": {"A": "No solution", "B": "20°", "C": "30°", "D": "40°"}, "correct_answer": "A"}, {"id": 22, "difficulty": "medium", "question": "Cyclic quadrilateral has sides 5,6,7,8 in order. Product of diagonals if area maximum?", "options": {"A": "√(17100)", "B": "√(18100)", "C": "√(19100)", "D": "√(20100)"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "In cyclic quadrilateral, diagonals are perpendicular. If diagonals lengths 12cm and 16cm, area?", "options": {"A": "48 cm²", "B": "64 cm²", "C": "96 cm²", "D": "192 cm²"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "ABCD cyclic with AB=6, BC=7, CD=8, DA=9. Using Ptolemy, AC×BD=?", "options": {"A": "117", "B": "118", "C": "119", "D": "120"}, "correct_answer": "C"},
    {"id": 25, "difficulty": "medium", "question": "In cyclic quadrilateral, angles A,B,C,D in ratio 2:3:4:3. Angles?", "options": {"A": "60°,90°,120°,90°", "B": "50°,75°,100°,75°", "C": "40°,60°,80°,60°", "D": "45°,67.5°,90°,67.5°"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Cyclic quadrilateral circumscribed about circle (tangential). Sum of pairs opposite sides?", "options": {"A": "Equal", "B": "In ratio 2:1", "C": "Product constant", "D": "Different"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Isosceles trapezium inscribed in circle. If parallel sides 10cm and 18cm, area?", "options": {"A": "84√2 cm²", "B": "96√2 cm²", "C": "108√2 cm²", "D": "120√2 cm²"}, "correct_answer": "B"},
   {"id": 28, "difficulty": "medium", "question": "In cyclic quadrilateral ABCD, ∠A=60°, ∠B=80°. Exterior angle at C equals?", "options": {"A": "60°", "B": "80°", "C": "100°", "D": "120°"}, "correct_answer": "A"}, {"id": 29, "difficulty": "medium", "question": "Circle passes through vertices of parallelogram. It must be:", "options": {"A": "Rectangle", "B": "Rhombus", "C": "Square", "D": "Any parallelogram"}, "correct_answer": "A"},
   {"id": 30, "difficulty": "medium", "question": "Cyclic quadrilateral with sides 3,3,4,4. Radius of circumcircle?", "options": {"A": "3", "B": "√10", "C": "√13", "D": "√15"}, "correct_answer": "A"}, {"id": 31, "difficulty": "medium", "question": "In cyclic quadrilateral, diagonals intersect at right angles. Sides a,b,c,d in order. Relation?", "options": {"A": "a²+c²=b²+d²", "B": "a²+b²=c²+d²", "C": "ab=cd", "D": "a/b=c/d"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "ABCD cyclic with ∠A=70°, ∠B=110°, ∠C=70°, ∠D=110°. Type?", "options": {"A": "Rectangle", "B": "Isosceles trapezium", "C": "Parallelogram", "D": "Kite"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "Using Ptolemy, find diagonal of cyclic quadrilateral sides 5,6,7,8 if other diagonal 10.", "options": {"A": "9.4", "B": "9.6", "C": "9.8", "D": "10.0"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "In cyclic quadrilateral, sum of squares of sides equals:", "options": {"A": "Sum of squares of diagonals", "B": "Product of diagonals", "C": "8R²", "D": "Depends on angles"}, "correct_answer": "D"},
    {"id": 35, "difficulty": "medium", "question": "Quadrilateral with angles 100°,80°,100°,80° is cyclic?", "options": {"A": "Yes", "B": "No", "C": "Can't say", "D": "If sides equal"}, "correct_answer": "A"},
   {"id": 36, "difficulty": "medium", "question": "Maximum area cyclic quadrilateral with sides a,b,c,d given by Brahmagupta formula √((s-a)(s-b)(s-c)(s-d)). For sides 6,7,8,9, area?", "options": {"A": "√3024", "B": "12√21", "C": "6√84", "D": "All represent same area"}, "correct_answer": "D"}, {"id": 37, "difficulty": "medium", "question": "In circle, chords AB and CD intersect at E. If AE=4, EB=6, CE=3, ED=8, quadrilateral ACBD cyclic?", "options": {"A": "Yes", "B": "No", "C": "Can't say", "D": "If angles known"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Cyclic quadrilateral with perpendicular diagonals. Area=½d1d2. For diagonals 10 and 24, circumradius?", "options": {"A": "12", "B": "13", "C": "14", "D": "15"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "If quadrilateral is both cyclic and tangential, sum of opposite sides:", "options": {"A": "Equal", "B": "Product constant", "C": "In ratio 2:1", "D": "Different"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "In cyclic quadrilateral, external angle at A equals 110°. Internal ∠C=?", "options": {"A": "70°", "B": "110°", "C": "120°", "D": "130°"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "Ptolemy for cyclic quadrilateral ABCD: AC×BD = AB×CD + AD×BC. If AB=BC=CD=DA, quadrilateral is:", "options": {"A": "Square", "B": "Rhombus", "C": "Rectangle", "D": "Any"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "Cyclic quadrilateral with sides 7,8,9,10. Semipermimeter s=?", "options": {"A": "16", "B": "17", "C": "18", "D": "19"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "In cyclic quadrilateral, angle between diagonals equals angle between opposite sides?", "options": {"A": "Always", "B": "Sometimes", "C": "Never", "D": "If perpendicular"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "Quadrilateral with vertices on circle and one diagonal as diameter. Then:", "options": {"A": "Opposite angles supplementary", "B": "Angles in semicircle right", "C": "Both A and B", "D": "Neither"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "medium", "question": "If cyclic quadrilateral has sides 5,6,7,8, maximum area occurs when?", "options": {"A": "It is cyclic", "B": "It is tangential", "C": "Diagonals perpendicular", "D": "It is rectangle"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "In cyclic quadrilateral ABCD, ∠A:∠B:∠C=4:5:6. ∠D=?", "options": {"A": "90°", "B": "100°", "C": "110°", "D": "120°"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Two circles intersect at A,B. Lines through A,B meet circles at C,D,E,F. Quadrilateral CDEF is:", "options": {"A": "Cyclic", "B": "Parallelogram", "C": "Trapezium", "D": "Not cyclic"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "Cyclic quadrilateral with diagonals as diameters of circle must be:", "options": {"A": "Rectangle", "B": "Square", "C": "Rhombus", "D": "Kite"}, "correct_answer": "A"},
   {"id": 49, "difficulty": "medium", "question": "Using Brahmagupta, area cyclic quadrilateral sides 13,14,15,16?", "options": {"A": "√(16×15×14×13)", "B": "√(29×16×15×14)", "C": "√(43680)", "D": "All of these"}, "correct_answer": "D"}, {"id": 50, "difficulty": "medium", "question": "In cyclic quadrilateral, if one diagonal bisects other, quadrilateral is?", "options": {"A": "Parallelogram", "B": "Trapezium", "C": "Kite", "D": "Not determined"}, "correct_answer": "D"},
  {"id": 51, "difficulty": "medium", "question": "ABCD cyclic with AB=5, BC=12, CD=18, DA=11. AC×BD from Ptolemy?", "options": {"A": "5×18+12×11=222", "B": "5×12+18×11=258", "C": "5×11+12×18=271", "D": "5×18+11×12=222"}, "correct_answer": "A"}, {"id": 52, "difficulty": "medium", "question": "Cyclic quadrilateral maximum area for given sides occurs when:", "options": {"A": "It is cyclic", "B": "It is also tangential", "C": "Diagonals perpendicular", "D": "Angles equal"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "Quadrilateral with angles 85°,95°,85°,95° cyclic?", "options": {"A": "Yes", "B": "No", "C": "If sides equal", "D": "Can't say"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "In cyclic quadrilateral, product of diagonals equals sum products opposite sides. This is:", "options": {"A": "Ptolemy's theorem", "B": "Brahmagupta theorem", "C": "Apollonius theorem", "D": "Euler's theorem"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "Cyclic trapezium with bases 8 and 18, legs 13 each. Height?", "options": {"A": "12", "B": "13", "C": "14", "D": "15"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "For quadrilateral to be cyclic, sufficient condition:", "options": {"A": "Opposite angles supplementary", "B": "Exterior angle equals interior opposite", "C": "Both", "D": "Either"}, "correct_answer": "D"},
    {"id": 57, "difficulty": "medium", "question": "In circle, quadrilateral with three right angles is:", "options": {"A": "Rectangle", "B": "Square", "C": "Trapezium", "D": "Cyclic"}, "correct_answer": "A"},
   {"id": 58, "difficulty": "medium", "question": "Cyclic quadrilateral with sides 8,8,15,15. Area?", "options": {"A": "120", "B": "15×8", "C": "√(15²×8²)", "D": "All of these"}, "correct_answer": "D"}, {"id": 59, "difficulty": "medium", "question": "In cyclic quadrilateral ABCD, AB∥CD. Then it is:", "options": {"A": "Parallelogram", "B": "Isosceles trapezium", "C": "Rectangle", "D": "Rhombus"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "Using Ptolemy, diagonal of cyclic quadrilateral sides 7,8,9,10 if other diagonal √(171).", "options": {"A": "√(154)", "B": "√(164)", "C": "√(174)", "D": "√(184)"}, "correct_answer": "C"},
  {"id": 61, "difficulty": "hard", "question": "Cyclic quadrilateral ABCD has AB=3, BC=4, CD=5, DA=6. Find AC×BD.", "options": {"A": "3×5+6×4=39", "B": "3×4+5×6=42", "C": "3×6+4×5=38", "D": "3×5+4×6=39"}, "correct_answer": "A"},  {"id": 62, "difficulty": "hard", "question": "If cyclic quadrilateral also has inscribed circle, sum opposite sides equal. For sides a,b,c,d, r=?", "options": {"A": "√((s-a)(s-b)(s-c)(s-d))/s", "B": "√(abcd)/s", "C": "√((ab+cd)(ad+bc)/(ab+cd+ad+bc))", "D": "All of these"}, "correct_answer": "D"},
    {"id": 63, "difficulty": "hard", "question": "In cyclic quadrilateral, diagonals d1,d2, sides a,b,c,d. Ptolemy gives d1d2=ac+bd. For maximum area, d1,d2 satisfy?", "options": {"A": "d1²+d2²=a²+b²+c²+d²", "B": "d1/d2=√(ac/bd)", "C": "d1²d2²=(ac+bd)²", "D": "No relation"}, "correct_answer": "C"},
    {"id": 64, "difficulty": "hard", "question": "Cyclic quadrilateral with perpendicular diagonals. Area=½d1d2. If sides a,b,c,d, then a²+c²=b²+d². Prove.", "options": {"A": "Using Ptolemy and diagonal lengths", "B": "Using cosine rule", "C": "Both", "D": "Neither proves"}, "correct_answer": "C"},
   {"id": 65, "difficulty": "hard", "question": "Quadrilateral both cyclic and tangential. If sides a,b,c,d, then area=√(abcd). This is:", "options": {"A": "Special case of Brahmagupta", "B": "Ptolemy's theorem", "C": "Parameshvara's formula", "D": "Bretschneider's formula"}, "correct_answer": "A"},{"id": 65, "difficulty": "hard", "question": "Quadrilateral both cyclic and tangential. If sides a,b,c,d, then area=√(abcd). This is:", "options": {"A": "Brahmagupta formula", "B": "Ptolemy's theorem", "C": "Parameshvara's formula", "D": "Bretschneider's formula"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "hard", "question": "In cyclic quadrilateral, angles satisfy sin(A/2)sin(C/2)=sin(B/2)sin(D/2). This follows from:", "options": {"A": "Law of sines", "B": "Ptolemy", "C": "Both", "D": "Neither"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "hard", "question": "Maximum area quadrilateral with sides a,b,c,d is cyclic. For given perimeter, maximum area when?", "options": {"A": "Cyclic", "B": "Square", "C": "Both", "D": "Neither"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "hard", "question": "Cyclic quadrilateral with sides a,b,c,d has circumradius R=¼√((ab+cd)(ac+bd)(ad+bc))/(Area). For a=5,b=6,c=7,d=8, R≈?", "options": {"A": "4.5", "B": "5.0", "C": "5.5", "D": "6.0"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "hard", "question": "In quadrilateral, if sum of products of opposite sides equals product of diagonals, it is cyclic. This is:", "options": {"A": "Converse of Ptolemy", "B": "Ptolemy's inequality", "C": "Both", "D": "Neither"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "hard", "question": "Cyclic quadrilateral ABCD with diagonals meeting at E. Then AE×EC=BE×ED if:", "options": {"A": "Always", "B": "Never", "C": "Only if cyclic", "D": "Only if tangential"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "hard", "question": "For quadrilateral to be cyclic, necessary and sufficient:", "options": {"A": "Opposite angles supplementary", "B": "Ptolemy holds", "C": "Both", "D": "Either"}, "correct_answer": "C"},
    {"id": 72, "difficulty": "hard", "question": "In cyclic quadrilateral, if one side is diameter, opposite angle is right. If two sides are diameters, quadrilateral is:", "options": {"A": "Rectangle", "B": "Square", "C": "Both", "D": "Neither"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "hard", "question": "Cyclic quadrilateral with sides 6,7,8,9. Using Brahmagupta, area=√(15×9×8×7) but s=15 not 15. Correct s?", "options": {"A": "15", "B": "16", "C": "17", "D": "18"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "hard", "question": "If quadrilateral has perpendicular diagonals, and vertices concyclic, then midpoints of sides lie on circle with center at intersection of diagonals?", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "B"},
    {"id": 75, "difficulty": "hard", "question": "Cyclic quadrilateral maximum area for given sides a,b,c,d is √((s-a)(s-b)(s-c)(s-d)). For a=7,b=8,c=9,d=10, s=17, area?", "options": {"A": "√(10×9×8×7)=√5040", "B": "√(10×9×8×7)=√5040≈71", "C": "√(10×9×8×7)=√5040≈71", "D": "All same"}, "correct_answer": "D"},
    {"id": 76, "difficulty": "hard", "question": "In cyclic quadrilateral, external angle at vertex equals internal opposite angle. If external angle 115°, opposite internal?", "options": {"A": "65°", "B": "115°", "C": "125°", "D": "245°"}, "correct_answer": "B"},
    {"id": 77, "difficulty": "hard", "question": "Quadrilateral with vertices on circle and diagonals perpendicular. Area formula?", "options": {"A": "½d1d2", "B": "√(s-a)(s-b)(s-c)(s-d)", "C": "Both", "D": "Neither"}, "correct_answer": "C"},
    {"id": 78, "difficulty": "hard", "question": "Ptolemy's theorem extended: For any quadrilateral, AC×BD ≤ AB×CD + AD×BC with equality iff cyclic. This is:", "options": {"A": "Ptolemy's inequality", "B": "Ptolemy's theorem", "C": "Converse", "D": "Generalization"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "hard", "question": "In cyclic quadrilateral, if diagonals are angle bisectors, quadrilateral is:", "options": {"A": "Rhombus", "B": "Kite", "C": "Isosceles trapezium", "D": "Not determined"}, "correct_answer": "D"},
    {"id": 80, "difficulty": "hard", "question": "Cyclic quadrilateral with sides a,a,b,b. Area using Brahmagupta?", "options": {"A": "√((a+b)²(a-b)²)/4", "B": "ab", "C": "√(ab(a+b)²)", "D": "√(a²b²)"}, "correct_answer": "B"},
    {"id": 81, "difficulty": "hard", "question": "In circle, chords AB and CD intersect at E. Quadrilateral ACBD cyclic. If AE=3, EB=4, CE=6, ED=2, using intersecting chords, verify cyclic.", "options": {"A": "3×4=6×2? 12=12 yes", "B": "3×6=4×2? 18=8 no", "C": "3×2=4×6? 6=24 no", "D": "None"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "For cyclic quadrilateral, sum of squares of sides equals sum of squares of diagonals plus four times square of circumradius?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Never"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "In cyclic quadrilateral, if one diagonal is diameter, and other diagonal length d, sides a,b,c,d in order, relation?", "options": {"A": "a²+b²=c²+d²", "B": "a²+c²=b²+d²", "C": "ab=cd", "D": "a/b=c/d"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Quadrilateral with sides 5,5,5,5 and diagonals 6,8 cyclic?", "options": {"A": "Yes", "B": "No", "C": "Can't say", "D": "If angles 90°"}, "correct_answer": "B"},
    {"id": 85, "difficulty": "hard", "question": "Cyclic quadrilateral with perpendicular diagonals. If sides are 6,7,8,9, area using both formulas equal?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "Using Ptolemy, find diagonal of cyclic pentagon?", "options": {"A": "Not applicable", "B": "Extended Ptolemy", "C": "Similar formula", "D": "No formula"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "In cyclic quadrilateral, angle between extensions of opposite sides equals half difference of arcs?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "For quadrilateral inscribed in circle, product of diagonals equals sum of products of opposite sides. If sides equal, diagonals?", "options": {"A": "Equal", "B": "Perpendicular", "C": "Both", "D": "Neither"}, "correct_answer": "A"},
   {"id": 89, "difficulty": "hard", "question": "Cyclic quadrilateral with sides 1,2,3,4. Maximum area?", "options": {"A": "√10", "B": "√15", "C": "√20", "D": "√24"}, "correct_answer": "D"}, {"id": 90, "difficulty": "hard", "question": "In quadrilateral, if opposite angles supplementary, then exists circle through all vertices. Proof uses:", "options": {"A": "Contradiction", "B": "Construction", "C": "Both", "D": "Neither"}, "correct_answer": "B"},
    {"id": 91, "difficulty": "hard", "question": "Cyclic quadrilateral ABCD with AB=AD and CB=CD. It is:", "options": {"A": "Kite", "B": "Rhombus", "C": "Square", "D": "Deltoid"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "Ptolemy for cyclic quadrilateral: AC×BD=AB×CD+AD×BC. For rectangle sides a,b, diagonals √(a²+b²), verify.", "options": {"A": "√(a²+b²)×√(a²+b²)=a×a+b×b", "B": "√(a²+b²)×√(a²+b²)=a×b+b×a", "C": "Both wrong", "D": "Both correct"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "In cyclic quadrilateral, if one angle is 60° and opposite 120°, sides ratio?", "options": {"A": "1:√3:2:√3", "B": "1:2:1:2", "C": "Depends", "D": "No fixed ratio"}, "correct_answer": "D"},
    {"id": 94, "difficulty": "hard", "question": "Quadrilateral with vertices (0,0),(a,0),(a,b),(0,b) cyclic?", "options": {"A": "Yes", "B": "No", "C": "If a=b", "D": "If rectangle"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Using Brahmagupta, area cyclic quadrilateral sides √5,√6,√7,√8. s=(√5+√6+√7+√8)/2. Area?", "options": {"A": "√((s-√5)(s-√6)(s-√7)(s-√8))", "B": "Simplify?", "C": "Approximately 3.5", "D": "All of these"}, "correct_answer": "D"},
    {"id": 96, "difficulty": "hard", "question": "In cyclic quadrilateral, perpendiculars from midpoints of sides to opposite sides concur at anticenter. Anticenter is:", "options": {"A": "Reflection of circumcenter in centroid", "B": "Midpoint of line joining circumcenter to orthocenter", "C": "Both", "D": "Neither"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Cyclic quadrilateral with given sides has maximum area when also tangential?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Quadrilateral with sides 3,4,5,6. Maximum possible area if cyclic?", "options": {"A": "√(9×5×4×2)=√360", "B": "√(9×5×4×2)=6√10", "C": "≈18.97", "D": "All of these"}, "correct_answer": "D"},
    {"id": 99, "difficulty": "hard", "question": "In circle, chords AB,CD,EF concur. Hexagon ACEBDF cyclic?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Ptolemy's theorem for cyclic quadrilateral ABCD: AC×BD=AB×CD+AD×BC. If AB=BC=CD=DA, then AC=BD√2?", "options": {"A": "Yes", "B": "No", "C": "If square", "D": "If rhombus"}, "correct_answer": "C"}
  ]
}