Fun Geometry Problem with Solution #150
⨷ 1



˹ x < 35°
٨ x= 30°
٨ 1



(1)∠BCD = 50°- x Р∠ADC = 70°- x

(2) AC ͡ѧش P · CP = DP ∠BCP = 20°+ x
∵ CP = DP ∆CDP 繠∆˹Ҩ ՠ∠P ʹ ∠CDP =∠DCP ∠CDP = 70° ∠CPD = 40°
ѧࡵ CP = DP Р∠CPD = 2(∠CBD) ش P circumcenter ͧ∆BCD BP = CP = DP

BP = L CP = L DP = L

(3) ˹ش Q AD AQ = L Р∠DAQ = 20°+ x
Ҡ∆ADQ∆BCP ¤ѹẺ -- (AD = BC,∠DAQ =∠BCP, AQ = CP) DQ = BP (= L) DQ = AQ ∆ADQ 繠∆˹Ҩ ՠ∠Q ʹ ∠ADQ =∠DAQ ∠ADQ = 20°+ x ∠AQD = 140°- 2x

(4) ∵DP = DQ ∆DPQ 繠∆˹Ҩ ՠ∠D (= 160°) ʹ ∠DPQ = 10°Р∠DQP = 10° ∠APQ = 30°Р∠AQP = 130°- 2x (> 60° x < 35°)

(5) ˹ش R Ҿз͹ͧش Q ҹ AP ∆APR∆APQ PR = PQ, AR = AQ = L ∠APR =∠APQ = 30°
∵ PQ = PR Р∠QPR = 60° ∆PQR 繠∆ҹ PQ = QR
͡ҡ ѧҠ∠PQR = 60° ∠AQR = 70°- 2x

(6) ѧࡵҠ∆AQR∆DPQ ¤ѹẺ -- (AQ = DP, AR = DQ, QR = PQ) ∠AQR =∠DPQ 70°- 2x = 10° x = 30° Q.E.D.

٨ 2



(1)∠BCD = 50°- x

(2) AC ͡ѧش P · CP = DP ∠BCP = 20°+ x
∵ CP = DP ∆CDP 繠∆˹Ҩ ՠ∠P ʹ ∠CDP =∠DCP ∠CDP = 70° ∠CPD = 40°
ѧࡵ CP = DP Р∠CPD = 2(∠CBD) ش P circumcenter ͧ∆BCD BP = CP = DP

BP = L CP = L DP = L

(3) ˹ش Q AD DQ = L Р∠ADQ = 20°+ x
Ҡ∆ADQ∆BCP ¤ѹẺ -- (AD = BC,∠ADQ =∠BCP, DQ = CP) AQ = BP AQ = L

(4) ˹ش R AP DR = L DR = DP ∆DPR 繠∆˹Ҩ ՠ∠D ʹ ∠DRP =∠DPR ∠DRP = 40° ∠ADR = 40°- x

(5)∵DQ = DR Р∠QDR = 60° ∆DQR 繠∆ҹ QR = L Р∠DQR = 60°
∵ AQ = DQ = QR ش Q circumcenter ͧ∆ADR ∠DAR = (∠DQR)/2 x = 30° Q.E.D.



⨷ 2



˹ x > 35°
٨ x= 40°
٨



(1)∠BCD = 50°- x Р∠ADC = 70°- x

(2) AC ͡ѧش P · CP = DP ∠BCP = 20°+ x
∵ CP = DP ∆CDP 繠∆˹Ҩ ՠ∠P ʹ ∠CDP =∠DCP ∠CDP = 70° ∠CPD = 40°
ѧࡵ CP = DP Р∠CPD = 2(∠CBD) ش P circumcenter ͧ∆BCD BP = CP = DP

BP = L CP = L DP = L

(3) ˹ش Q AD AQ = L Р∠DAQ = 20°+ x
Ҡ∆ADQ∆BCP ¤ѹẺ -- (AD = BC,∠DAQ =∠BCP, AQ = CP) DQ = BP (= L) DQ = AQ ∆ADQ 繠∆˹Ҩ ՠ∠Q ʹ ∠ADQ =∠DAQ ∠ADQ = 20°+ x ∠AQD = 140°- 2x

(4) ∵DP = DQ ∆DPQ 繠∆˹Ҩ ՠ∠D (= 160°) ʹ ∠DPQ = 10°Р∠DQP = 10° ∠APQ = 30°Р∠AQP = 130°- 2x (< 60° x > 35°)

(5) ˹ش R Ҿз͹ͧش Q ҹ AP ∆APR∆APQ PR = PQ, AR = AQ = L ∠APR =∠APQ = 30°
∵ PQ = PR Р∠QPR = 60° ∆PQR 繠∆ҹ PQ = QR
͡ҡ ѧҠ∠PQR = 60° ∠AQR = 2x - 70°

(6) ѧࡵҠ∆AQR∆DPQ ¤ѹẺ -- (AQ = DP, AR = DQ, QR = PQ) ∠AQR =∠DPQ 2x - 70° = 10° x = 40° Q.E.D.




Create Date : 15 áҤ 2558
Last Update : 15 áҤ 2558 0:00:01 .
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