Miscellaneous Problems
⨷Âì 1-1



¨§¾ÔÊÙ¨¹ìÇèÒ AB = CD
¾ÔÊÙ¨¹ì



(1) ∠ACD = 40° áÅР∠BCD = 10°

(2) ¡Ó˹´¨Ø´ P à»ç¹ÀÒ¾Êзé͹¢Í§¨Ø´ A ¼èÒ¹ BC   ⇒   ∆BCP ≅ ∆ABC   ⇒   BP = AB, ∠CBP = ∠ABC = 30° áÅР∠BCP = ∠ACB = 50°
∵ AB = BP áÅР∠ABP = 60°   ⇒   ∆ABP à»ç¹ ∆´éÒ¹à·èÒ   ⇒   AB = AP
¹Í¡¨Ò¡¹Ñé¹ Âѧä´éÇèÒ ∠BAP = 60°   ⇔   ∠CAP = 40°

(3) ÊѧࡵÇèÒ ∆ACP ≅ ∆ACD ´éǤÇÒÁÊÑÁ¾Ñ¹¸ìẺ Á-´-Á (∠CAP = ∠ACD, AC = AC, ∠ACP = ∠CAD)   ⇒   AP = CD   ⇔   AB = CD   Q.E.D.

⨷Âì 1-2



¨§¾ÔÊÙ¨¹ìÇèÒ AC = BD
¾ÔÊÙ¨¹ì 1



(1) ∠ACD = 60°

(2) µèÍ CD ÍÍ¡ä»Âѧ¨Ø´ P â´Â·Õè CP = AC   ⇒   ∠BDP = ∠ADC   ⇔   ∠BDP = 80°
∵ AC = CP áÅР∠ACP = 60°   ⇒   ∆ACP à»ç¹ ∆´éÒ¹à·èÒ   ⇔   AC = AP = CP
¹Í¡¨Ò¡¹Ñé¹ Âѧä´éÇèÒ ∠APC = 60°

(3) ÊѧࡵÇèÒ AP = CP áÅР∠APC = 2(∠ABC)   ⇒   ¨Ø´ P à»ç¹ circumcenter ¢Í§ ∆ABC   ⇒   BP = AP   ⇔   BP = AC
¹Í¡¨Ò¡¹Ñé¹ Âѧä´éÇèÒ ∠BPC = 2(∠BAC)   ⇔   ∠BPC = 80°   ⇔   ∠BPD = ∠BDP   ⇔   ∆BDP à»ç¹ ∆˹éÒ¨ÑèÇ ·ÕèÁÕ ∠B à»ç¹ÁØÁÂÍ´   ⇔   BP = BD   ⇔   AC = BD   Q.E.D.

¾ÔÊÙ¨¹ì 2



(1) ∠BDC = 100°

(2) ¡Ó˹´¨Ø´ P º¹ AB ·Õè·ÓãËé ∠APC = 40°   ⇔   ∠APC = ∠CAP   ⇔   ∆ACP à»ç¹ ∆˹éÒ¨ÑèÇ ·ÕèÁÕ ∠C à»ç¹ÁØÁÂÍ´   ⇔   AC = CP

(3) ¾Ô¨ÒóҠ∆BCD ¨ÐàËç¹ÇèÒ ∠B = 30°, ∠D = 100° áÅÐÁըش P º¹ BD ·Õè·ÓãËé ∠CPD = 40°   ⇒   CP = BD (´ÙÇÔ¸Õ¾ÔÊÙ¨¹ìã¹â¨·Âì 1-1)   ⇔   AC = BD   Q.E.D.


⨷Âì 2



¨§¾ÔÊÙ¨¹ìÇèÒ AD = BC
¾ÔÊÙ¨¹ì



(1) ∠BCD = 30° áÅР∠ADC = 126°

(2) ¡Ó˹´¨Ø´ O à»ç¹ circumcenter ¢Í§ ∆ABC   ⇔   AO = BO = CO
¹Í¡¨Ò¡¹Ñé¹ Âѧä´éÇèÒ ∠BOC = 2(∠BAC)   ⇔   ∠BOC = 60°
∵ BO = CO áÅР∠BOC = 60°   ⇒   ∆BCO à»ç¹ ∆´éÒ¹à·èÒ   ⇒   ...
     • BC = BO = CO   ⇔   BC = AO
     • ∠CBO = 60°   ⇔   ∠ABO = 36°
     • ∠BCO = 60°   ⇔   ∠DCO = 30°
∵ AO = BO   ⇔   ∆ABO à»ç¹ ∆˹éÒ¨ÑèÇ ·ÕèÁÕ ∠O à»ç¹ÁØÁÂÍ´   ⇔   ∠BAO = ∠ABO   ⇔   ∠BAO = 36°

(3) ÊѧࡵÇèÒ ∆CDO ≅ ∆BCD ´éǤÇÒÁÊÑÁ¾Ñ¹¸ìẺ ´-Á-´ (CD = CD, ∠DCO = ∠BCD, CO = BC)   ⇒   ∠CDO = ∠BDC   ⇔   ∠CDO = 54°   ⇔   ∠ADO = 72°

(4) ¾Ô¨ÒóҠ∆ADO ¨Ðä´éÇèÒ ∠AOD = 72°   ⇔   ∠AOD = ∠ADO   ⇔   ∆ADO à»ç¹ ∆˹éÒ¨ÑèÇ ·ÕèÁÕ ∠A à»ç¹ÁØÁÂÍ´   ⇔   AD = AO   ⇔   AD = BC   Q.E.D.

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Create Date : 25 àÁÉÒ¹ 2557
Last Update : 15 ÁԶعÒ¹ 2558 22:40:00 ¹.
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Contents
Problem 001 : quadrilateral, diagonal, 24°, 48°, 66°
Problem 002 : triangle, interior point, x, x, 2x, 3x
Problem 003 : triangle, interior point, 10°, 10°, 30°, 100°
Problem 004 : triangle, cevian, 2x, 3x
Problem 005 : triangle, cevian, x, 2x, 2x
Problem 006 : triangle, interior point, 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, interior point, 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, interior point, 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, interior point, 10°, 20°, 40°, 50°
Problem 030 : triangle, interior point, 90°, 2x, 3x, 5x
Problem 031 : triangle, interior point, 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, interior point, 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, interior point, x, 2x, 3x, 4x, 5x
Problem 057 : triangle, cevian, α, β, α + β
Problem 058 : triangle, interior point, 20°, 30°, 30°, x, 2x
Problem 059 : triangle, cevian, 30°, 45°
Problem 060 : square, interior point, 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, interior point, 10°, 20°, 20°, 30°
Problem 068 : triangle, cevian, 15°, 30°
Problem 069 : quadrilateral, diagonal, α, 2α, β, 2β
Problem 070 : triangle, interior point, 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, interior point, 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, interior point, x, 3x, 5x
Problem 085 : quadrilateral, diagonal, 20°, 30°, 40°, 130°
Problem 086 : triangle, 18°, 30°
Problem 087 : triangle, interior point, perpendicular, 45°
Problem 088 : triangle, interior point, 10°, 10°, 10°, 30°
Problem 089 : quadrilateral, diagonal, 40°, 70°
Problem 090 : triangle, interior point, 10°, 20°, 30°, 40°
Problem 091 : square, interior point, 15°, 15°
Problem 092 : trigonometric equation 1
Problem 093 : triangle, cevian, 18°, 30°
Problem 094 : triangle, interior point, perpendicular, 30°, α, 30° - α
Problem 095 : triangle, cevian, 100°
Problem 096 : triangle, interior point, 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, interior point, 6°, 12°, 24°, 30°
Problem 101 : triangle, interior point, 20°, 60°, 120°
Problem 102 : triangle, cevian, x, 2x, 5x
Problem 103 : triangle, interior point, 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, interior point, 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, interior point, 10°, 20°, 40°, 70°
Problem 112 : triangle, interior point, 10°, 20°, 30°, 80°
Problem 113 : triangle, interior point, 90°, x, 2x, 2x
Problem 114 : triangle, interior point, 6°, 12°, 48°, 54°
Problem 115 : triangle, interior point, 6°, 30°
Problem 116 : quadrilateral, diagonal, 20°, 30°, 40°, 60°
Problem 117 : triangle, interior point, 6°, 24°, 48°, 66°
Problem 118 : triangle, interior point, 10°, 10°, 20°, 100°
Problem 119 : equilateral triangle, exterior point, 60°
Problem 120 : triangle, cevian, 40°, 50°
Problem 121 : quadrilateral, diagonal, 90°, 90°
Problem 122 : triangle, interior point, 2x, 3x, 3x, 4x
Problem 123 : trigonometric equation 2
Problem 124 : triangle, interior point, 10°, 10°, 20°, 40°
Problem 125 : triangle, interior point, 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, interior point, 12°, 18°, 36°, 36°
Problem 132 : triangle, interior point, 30°, α, 30° - α, 60° - α
Problem 133 : triangle, interior point, 10°, 20°, 30°, 40°
Problem 134 : triangle, interior point, 10°, 20°, 30°, 40°
Problem 135 : triangle, cevian, 10°, 80°
Problem 136 : triangle, 30°, 54°
Problem 137 : triangle, interior point, 5°, 10°, 20°, 20°
Problem 138 : quadrilateral, diagonal, 36°, 60°, 108°
Problem 139 : triangle, cevian, 12°, 30°
Problem 140 : triangle, interior point, 20°, 30°, 40°, 50°
Problem 141 : quadrilateral, diagonal, 60°, 60°, α, α
Problem 142 : triangle, interior point, 10°, 10°, 20°, 100°
Problem 143 : triangle, interior point, 20°, 20°, 30°, 40°
Problem 144 : quadrilateral, diagonal, 30°, 90°, x, x, 2x
Problem 145 : right triangle, α, β, γ
Problem 146 : triangle, interior point, 10°, 20°, 30°, 40°
Problem 147 : triangle, interior point, 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, interior point, 10°, 20°, 40°, 70°
Problem 153 : quadrilateral, diagonal, 20°, 20°, 50°, 100°
Problem 154 : triangle, interior point, 10°, 20°, 30°, 100°
Problem 155 : quadrilateral, diagonal, x, x, 2x, 2x, 3x
Problem 156 : quadrilateral, diagonal, 20°, 30°, 100°, 110°
Problem 157 : triangle, interior point, 10°, 10°, 30°, 70°
Problem 158 : quadrilateral, diagonal, 20°, 40°, 40°, 70°
Problem 159 : triangle, interior point, perpendicular, x, 2x, 3x, 5x
Problem 160 : triangle, interior point, 10°, 20°, 30°, 50°
Problem 161 : triangle, interior point, 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 àÁÉÒ¹ 2557
Last Update : 26 ÊÔ§ËÒ¤Á 2558 0:01:00 ¹.
Counter : 3896 Pageviews.
 
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