Miscellaneous Problems
⨷Âì 1-1
¨§¾ÔÊÙ¨¹ìÇèÒ AB = CD ¾ÔÊÙ¨¹ì
(1) ∠ACD = 40° áÅÐ ∠BCD = 10°
(2) ¡Ó˹´¨Ø´ P à»ç¹ÀÒ¾Êзé͹¢Í§¨Ø´ A ¼èÒ¹ BC ⇒ ∆BCP ≅ ∆ABC ⇒ BP = AB, ∠CBP = ∠ABC = 30° áÅÐ ∠BCP = ∠ACB = 50° ∵ AB = BP áÅÐ ∠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 áÅÐ ∠ACP = 60° ⇒ ∆ACP à»ç¹ ∆´éÒ¹à·èÒ ⇔ AC = AP = CP ¹Í¡¨Ò¡¹Ñé¹ Âѧä´éÇèÒ ∠APC = 60°
(3) ÊѧࡵÇèÒ AP = CP áÅÐ ∠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¨ÒÃ³Ò ∆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° áÅÐ ∠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 áÅÐ ∠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¨ÒÃ³Ò ∆ADO ¨Ðä´éÇèÒ ∠AOD = 72° ⇔ ∠AOD = ∠ADO ⇔ ∆ADO à»ç¹ ∆˹éÒ¨ÑèÇ ·ÕèÁÕ ∠A à»ç¹ÁØÁÂÍ´ ⇔ AD = AO ⇔ AD = BC Q.E.D.
Contents
Problem 003 : triangle, interior point, 10°, 10°, 30°, 100° Problem 013 : quadrilateral, diagonal, 13°, 13°, 30°, 86° Problem 019 : quadrilateral, diagonal, 20°, 20°, 80°, 100° Problem 023 : quadrilateral, diagonal, x, 2x, 2x, 4x, 4x Problem 027 : quadrilateral, diagonal, 30°, 33°, 76°, 98° Problem 029 : triangle, interior point, 10°, 20°, 40°, 50° Problem 030 : triangle, interior point, 90°, 2x, 3x, 5x Problem 036 : quadrilateral, diagonal, 27°, 38°, 54°, 63° Problem 037 : quadrilateral, diagonal, 20°, 30°, 30°, 50° Problem 044 : triangle, interior point, 10°, 20°, 20°, 30° Problem 050 : quadrilateral, diagonal, 12°, 30°, 42°, 72° Problem 056 : triangle, interior point, x, 2x, 3x, 4x, 5x Problem 058 : triangle, interior point, 20° , 30° , 30° , x, 2x Problem 067 : triangle, interior point, 10°, 20°, 20°, 30° Problem 073 : quadrilateral, diagonal, 10°, 20°, 40°, 100° Problem 074 : triangle, interior point, 10°, 20°, 30°, 40° Problem 076 : quadrilateral, diagonal, 30°, 60°, 60°, α Problem 080 : quadrilateral, diagonal, 30°, 40°, 60°, 70° Problem 084 : equilateral triangle, interior point, x, 3x, 5x Problem 085 : quadrilateral, diagonal, 20°, 30°, 40°, 130° Problem 087 : triangle, interior point, perpendicular, 45° Problem 088 : triangle, interior point, 10°, 10°, 10°, 30° Problem 090 : triangle, interior point, 10°, 20°, 30°, 40° Problem 094 : triangle, interior point, perpendicular, 30°, α, 30° - α Problem 096 : triangle, interior point, 6°, 12°, 24°, 30° Problem 100 : triangle, interior point, 6°, 12°, 24°, 30° Problem 103 : triangle, interior point, x, 2x, 2x, 2x, 3x Problem 107 : triangle, interior point, 6°, 24°, 30°, 48° Problem 108 : quadrilateral, diagonal, 20°, 30°, 40°, 70° Problem 111 : triangle, interior point, 10°, 20°, 40°, 70° Problem 112 : triangle, interior point, 10°, 20°, 30°, 80° Problem 114 : triangle, interior point, 6°, 12°, 48°, 54° 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 124 : triangle, interior point, 10°, 10°, 20°, 40° Problem 126 : quadrilateral, diagonal, 10°, 20°, 60°, 60° Problem 129 : quadrilateral, diagonal, 10°, 30°, 40°, 70° 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 137 : triangle, interior point, 5°, 10°, 20°, 20° Problem 140 : triangle, interior point, 20°, 30°, 40°, 50° 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 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 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 163 : quadrilateral, diagonal, 20°, 30°, 40°, 110° Problem 165 : quadrilateral, diagonal, 20°, 30°, 60°, 100°
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