Miscellaneous Problems
⨷ 1-1



٨ AB = CD
٨



(1)∠ACD = 40°Р∠BCD = 10°

(2) ˹ش P Ҿз͹ͧش A ҹ BC ∆BCP∆ABC BP = AB,∠CBP =∠ABC = 30°Р∠BCP =∠ACB = 50°
∵ AB = BP Р∠ABP = 60° ∆ABP 繠∆ҹ AB = AP
͡ҡ ѧҠ∠BAP = 60° ∠CAP = 40°

(3) ѧࡵҠ∆ACP∆ACD ¤ѹẺ -- (∠CAP =∠ACD, AC = AC,∠ACP =∠CAD) AP = CD AB = CD Q.E.D.

⨷ 1-2



٨ AC = BD
٨ 1



(1)∠ACD = 60°

(2) CD ͡ѧش P · CP = AC ∠BDP =∠ADC ∠BDP = 80°
∵ AC = CP Р∠ACP = 60° ∆ACP 繠∆ҹ AC = AP = CP
͡ҡ ѧ ∠APC = 60°

(3) ѧࡵ AP = CP Р∠APC = 2(∠ABC) ش P circumcenter ͧ∆ABC BP = AP BP = AC
͡ҡ ѧҠ∠BPC = 2(∠BAC) ∠BPC = 80° ∠BPD =∠BDP ∆BDP 繠∆˹Ҩ ՠ∠B ʹ BP = BD AC = BD Q.E.D.

٨ 2



(1)∠BDC = 100°

(2) ˹ش P AB ∠APC = 40° ∠APC =∠CAP ∆ACP 繠∆˹Ҩ ՠ∠C ʹ AC = CP

(3) ԨóҠ∆BCD Ҡ∠B = 30°,∠D = 100°ըش P BD ∠CPD = 40° CP = BD (Ըվ٨⨷ 1-1) AC = BD Q.E.D.



⨷ 2



٨ AD = BC
٨



(1)∠BCD = 30°Р∠ADC = 126°

(2) ˹ش O circumcenter ͧ∆ABC AO = BO = CO
͡ҡ ѧҠ∠BOC = 2(∠BAC) ∠BOC = 60°
∵ BO = CO Р∠BOC = 60° ∆BCO 繠∆ҹ ...
BC = BO = CO BC = AO
∠CBO = 60° ∠ABO = 36°
∠BCO = 60° ∠DCO = 30°
∵ AO = BO ∆ABO 繠∆˹Ҩ ՠ∠O ʹ ∠BAO =∠ABO ∠BAO = 36°

(3) ѧࡵҠ∆CDO∆BCD ¤ѹẺ -- (CD = CD,∠DCO =∠BCD, CO = BC) ∠CDO =∠BDC ∠CDO = 54° ∠ADO = 72°

(4) ԨóҠ∆ADO Ҡ∠AOD = 72° ∠AOD =∠ADO ∆ADO 繠∆˹Ҩ ՠ∠A ʹ AD = AO AD = BC Q.E.D.




Create Date : 25 ¹ 2557
Last Update : 15 Զع¹ 2558 22:40:00 .
Counter : 983 Pageviews.

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Contents
Problem 001 : quadrilateral, diagonal, 24°, 48°, 66°
Problem 002 : triangle,interiorpoint, x, x, 2x, 3x
Problem 003 :triangle,interiorpoint, 10°, 10°, 30°, 100°
Problem 004 :triangle, cevian, 2x, 3x
Problem 005 :triangle, cevian, x, 2x, 2x
Problem 006 :triangle,interiorpoint, 20°, 30°, 50°
Problem 007 : triangle, cevian, 20°, 80°
Problem 008 :triangle, cevian, 30°, 105°
Problem 009 :triangle, cevian,α, 2α
Problem 010 :triangle, cevian, 90°, x, 2x
Problem 011 :triangle, cevian, x, 2x, 90° + x
Problem 012 :quadrilateral, diagonal, 30°,α
Problem 013 :quadrilateral, diagonal, 13°, 13°, 30°, 86°
Problem 014 :quadrilateral,α, 2α
Problem 015 : triangle, inscribed circle
Problem 016 :triangle, cevian, 45°, x, 3x
Problem 017 :triangle, cevian, 45°, x, x
Problem 018 :triangle,interiorpoint, 90°, x, x
Problem 019 :quadrilateral, diagonal, 20°, 20°, 80°, 100°
Problem 020 :quadrilateral, diagonal, 48°, x, 5x, 5x
Problem 021 : triangle, a, b, c, x, 2x
Problem 022 :triangle,interiorpoint,α, 2α,β, 3β
Problem 023 :quadrilateral, diagonal, x, 2x, 2x, 4x, 4x
Problem 024 :triangle, cevian, 30°, x, 2x
Problem 025 :triangle, cevian, 15°, 50°
Problem 026 :triangle, cevian, 20°, 70°, 80°
Problem 027 :quadrilateral, diagonal, 30°, 33°, 76°, 98°
Problem 028 :triangle, cevian, 120°, 2x, 3x
Problem 029 :triangle,interiorpoint, 10°, 20°, 40°, 50°
Problem 030 :triangle,interiorpoint, 90°, 2x, 3x, 5x
Problem 031 :triangle,interiorpoint, 44°, 88°
Problem 032 :quadrilateral,α, 2α
Problem 033 :triangle, cevian, 24°, 30°
Problem 034 :triangle, cevian, x, 4x, 6x
Problem 035 :triangle, cevian, x, 2x
Problem 036 :quadrilateral, diagonal, 27°, 38°, 54°, 63°
Problem 037 : quadrilateral, diagonal, 20°, 30°, 30°, 50°
Problem 038 :triangle, cevian, 2x, 3x, 4x
Problem 039 :triangle, cevian, 12°, 18°
Problem 040 :quadrilateral, diagonal, 6°, 18°
Problem 041 :quadrilateral, diagonal, 70°, 80°, 140°
Problem 042 :triangle, cevian, 15°, 30°
Problem 043 :triangle, cevian, 20°, 60°, 70°
Problem 044 :triangle,interiorpoint, 10°, 20°, 20°, 30°
Problem 045 :triangle, cevian, x, 2x, 7x
Problem 046 :triangle, cevian, 90°, 2x, 3x
Problem 047 :triangle, cevian, x, 2x, 5x
Problem 048 :quadrilateral, diagonal, 30°, 3x, 4x
Problem 049 :triangle, cevian,α, 2α
Problem 050 :quadrilateral, diagonal, 12°, 30°, 42°, 72°
Problem 051 :triangle, cevian, 20°, 40°
Problem 052 :quadrilateral, diagonal, 30°,α, 3α
Problem 053 :triangle, cevian, 45°, x, 2x
Problem 054 :quadrilateral, diagonal, 90°
Problem 055 :triangle, 20°, 40°
Problem 056 :triangle,interiorpoint, x, 2x, 3x, 4x, 5x
Problem 057 :triangle, cevian,α, β,α+ β
Problem 058 :triangle,interiorpoint, 20°, 30°, 30°, x, 2x
Problem 059 :triangle, cevian, 30°, 45°
Problem 060 :square,interiorpoint, 9°, 36°
Problem 061 :triangle, cevian, 15°, 45°
Problem 062 :triangle, cevian, 24°, 54°
Problem 063 :quadrilateral, diagonal, 24°, 36°
Problem 064 :triangle, cevian, 42°, 84°
Problem 065 :triangle, cevian,x, 4x, 90° - x
Problem 066 : quadrilateral, 40°, 80°
Problem 067 :triangle,interiorpoint, 10°, 20°, 20°, 30°
Problem 068 :triangle, cevian, 15°, 30°
Problem 069 : quadrilateral, diagonal,α, 2α,β, 2β
Problem 070 :triangle,interiorpoint,α,β, γ, γ
Problem 071 :triangle, cevian,30°, x, 2x
Problem 072 :triangle, cevian,30°, 30°
Problem 073 :quadrilateral, diagonal, 10°, 20°, 40°, 100°
Problem 074 :triangle,interiorpoint, 10°, 20°, 30°, 40°
Problem 075 :triangle, incenter
Problem 076 : quadrilateral, diagonal, 30°, 60°, 60°,α
Problem 077 : square, parallel lines
Problem 078 :triangle, cevian, 3x, 4x, 5x
Problem 079 :triangle, cevian,30°, x, x
Problem 080 : quadrilateral, diagonal, 30°, 40°, 60°, 70°
Problem 081 : equilateral triangle,α
Problem 082 : regular nonagon, diagonal
Problem 083 :triangle, cevian,30°, 40°
Problem 084 : equilateral triangle,interiorpoint, x, 3x, 5x
Problem 085 :quadrilateral, diagonal, 20°, 30°, 40°, 130°
Problem 086 : triangle, 18°, 30°
Problem 087 :triangle,interiorpoint,perpendicular, 45°
Problem 088 :triangle,interiorpoint, 10°, 10°, 10°, 30°
Problem 089 : quadrilateral, diagonal, 40°, 70°
Problem 090 :triangle,interiorpoint, 10°, 20°, 30°, 40°
Problem 091 :square,interiorpoint, 15°, 15°
Problem 092 : trigonometric equation 1
Problem 093 :triangle, cevian, 18°, 30°
Problem 094 :triangle,interiorpoint,perpendicular, 30°,α, 30° -α
Problem 095 :triangle, cevian, 100°
Problem 096 :triangle,interiorpoint, 6°, 12°, 24°, 30°
Problem 097 : quadrilateral, diagonal, 90°, 105°
Problem 098 : 80-80-20 triangle (original problem)
Problem 099 :triangle, cevian, 18°, 48°
Problem 100 :triangle,interiorpoint, 6°, 12°, 24°, 30°
Problem 101 :triangle,interiorpoint, 20°, 60°, 120°
Problem 102 :triangle, cevian, x, 2x, 5x
Problem 103 :triangle,interiorpoint, x, 2x, 2x, 2x, 3x
Problem 104 : triangle, cevian, 30°, 135°
Problem 105 :triangle, cevian, 30°, 50°
Problem 106 : 80-80-20 triangle (60-70 variant)
Problem 107 :triangle,interiorpoint, 6°, 24°, 30°, 48°
Problem 108 : quadrilateral, diagonal, 20°, 30°, 40°, 70°
Problem 109 : triangle, angle bisector
Problem 110 :triangle, cevian, 2x, 2x, 90°+ x
Problem 111 :triangle,interiorpoint, 10°, 20°, 40°, 70°
Problem 112 :triangle,interiorpoint, 10°, 20°, 30°, 80°
Problem 113 :triangle,interiorpoint, 90°, x, 2x, 2x
Problem 114 :triangle,interiorpoint, 6°, 12°, 48°, 54°
Problem 115 :triangle,interiorpoint, 6°, 30°
Problem 116 : quadrilateral, diagonal, 20°, 30°, 40°, 60°
Problem 117 :triangle,interiorpoint, 6°, 24°, 48°, 66°
Problem 118 :triangle,interiorpoint, 10°, 10°, 20°, 100°
Problem 119 : equilateral triangle,exteriorpoint, 60°
Problem 120 :triangle, cevian, 40°, 50°
Problem 121 : quadrilateral, diagonal, 90°, 90°
Problem 122 :triangle,interiorpoint, 2x, 3x, 3x, 4x
Problem 123 :trigonometric equation 2
Problem 124 :triangle,interiorpoint, 10°, 10°, 20°, 40°
Problem 125 :triangle,interiorpoint, 30°, 40°
Problem 126 : quadrilateral, diagonal, 10°, 20°, 60°, 60°
Problem 127 :triangle, cevian,30°, 30°
Problem 128 : triangle, cevian, 60°,α,α
Problem 129 : quadrilateral, diagonal, 10°, 30°, 40°, 70°
Problem 130 : right triangle, midpoint
Problem 131 :triangle,interiorpoint, 12°, 18°, 36°, 36°
Problem 132 :triangle,interiorpoint, 30°,α, 30°-α, 60°-α
Problem 133 :triangle,interiorpoint, 10°, 20°, 30°, 40°
Problem 134 :triangle,interiorpoint, 10°, 20°, 30°, 40°
Problem 135 :triangle, cevian,10°, 80°
Problem 136 : triangle, 30°, 54°
Problem 137 :triangle,interiorpoint, 5°, 10°, 20°, 20°
Problem 138 : quadrilateral, diagonal, 36°, 60°, 108°
Problem 139 :triangle, cevian,12°, 30°
Problem 140 :triangle,interiorpoint, 20°, 30°, 40°, 50°
Problem 141 : quadrilateral, diagonal, 60°, 60°,α,α
Problem 142 :triangle,interiorpoint,10°, 10°, 20°, 100°
Problem 143 :triangle,interiorpoint,20°, 20°, 30°, 40°
Problem 144 : quadrilateral, diagonal, 30°, 90°, x, x, 2x
Problem 145 : right triangle,α,β
Problem 146 :triangle,interiorpoint, 10°, 20°, 30°, 40°
Problem 147 :triangle,interiorpoint, 10°, 20°, 60°, 60°
Problem 148 :quadrilateral, diagonal, 10°, 30°, 60°, 60°
Problem 149 : quadrilateral, diagonal, 20°, 30°, 30°, 80°
Problem 150 :triangle, cevian,20°, 110°
Problem 151 : triangle, cevian, perpendicular, x, x, 2x
Problem 152 :triangle,interiorpoint, 10°, 20°, 40°, 70°
Problem 153 : quadrilateral, diagonal, 20°, 20°, 50°, 100°
Problem 154 :triangle,interiorpoint, 10°, 20°, 30°, 100°
Problem 155 : quadrilateral, diagonal, x, x, 2x, 2x, 3x
Problem 156 : quadrilateral, diagonal, 20°, 30°, 100°, 110°
Problem 157 :triangle,interiorpoint, 10°, 10°, 30°, 70°
Problem 158 : quadrilateral, diagonal, 20°, 40°, 40°, 70°
Problem 159 :triangle,interiorpoint,perpendicular, x, 2x, 3x, 5x
Problem 160 :triangle,interiorpoint, 10°, 20°, 30°, 50°
Problem 161 :triangle,interiorpoint, 10°, 10°, 20°, 30°
Problem 162 : triangle, 60°, 80°
Problem 163 : quadrilateral, diagonal, 20°, 30°, 40°, 110°
Problem 164 : trigonometric equation 3
Problem 165 : quadrilateral, diagonal, 20°, 30°, 60°, 100°



Create Date : 22 ¹ 2557
Last Update : 26 ԧҤ 2558 0:01:00 .
Counter : 3365 Pageviews.

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