Fun Geometry Problem with Solution #100



٨ x = 6°
٨



(1) ∠ACB = 108°
∵ ∠BAC =∠ABC ∆ABC 繠∆˹Ҩ ՠ∠C ʹ AC = BC

(2) BP ͡ѧش D ·∠DAP = 24° ∠CAD = 6°,∠ADP = 126°Р∠APD = 30°
α= 6°
ѧࡵҠ☐ACBD AC = BC,∠A =α,∠B = 2αР∠C = 120°- 2α BC = BD (Click ʹԸվ٨⨷ 2) ∆BCD 繠∆˹Ҩ ՠ∠B (= 12°) ʹ ∠BCD (=∠BDC) = 84° ∠ACD = 24°

(3) CD ͡仾 AP ش E ∠ADE = 30° ∠EDP = 96°Р∠DEP = 54°

(4) ˹ش O 繨شٹҧͧǧՠ∆ADP Ṻ ...
AO = DO = OP
∠AOD = 2(∠APD) ∠AOD = 60°
∠DOP = 2(∠DAP) ∠DOP = 48°
∵ AO = DO Р∠AOD = 60° ∆ADO 繠∆ҹ ...
AD = AO AD = OP
∠DAO = 60° ∠OAP = 36°
∠ADO = 60° ∠EDO = 30°
ѧࡵҠ∆DEO∆ADE ¤ѹẺ -- (DO = AD,∠EDO = ∠ADE, DE = DE) ∠DOE =∠DAE ∠DOE = 24°
∵ AO = OP ∆AOP 繠∆˹Ҩ ՠ∠O ʹ ∠APO =∠OAP ∠APO = 36°
ԨóҠ∆EOP Ҡ∠OEP = 72° ∠OEP =∠EOP ∆EOP 繠∆˹Ҩ ՠ∠P ʹ EP = OP EP = AD

(5) DE = k
˹ش Q 繨شٹҧͧǧՠ∆DEP Ṻ ...
DQ = EQ = PQ
∠DQE = 2(∠DPE) ∠DQE = 60°
∠DQP = 2(∠DEP) ∠DQP = 108°
∵ DQ = EQ Р∠DQE = 60° ∆DEQ 繠∆ҹ EQ = DE EQ = k PQ = k
∵ EQ = PQ ∆EPQ 繠∆˹Ҩ ՠ∠Q (= 168°) ʹ ∠PEQ =∠EPQ = 6°

(6) ˹ش F AC AF = k
Ҡ∆ADF∆EPQ ¤ѹẺ -- (AD = EP,∠DAF =∠PEQ, AF = EQ) DF = PQ DF = k
͡ҡ ѧҠ∠AFD =∠EQP ∠AFD = 168° ∠CFD = 12°

(7) FC ͡ѧش G ∠DGF = 12° ∠DGF =∠DFG ∆DFG 繠∆˹Ҩ ՠ∠D ʹ DG = DF DG = k
͡ҡ ѧҠ∠DCG = 156° ∠CDG = 12°(∠GDP = 72°) ∠CDG =∠CGD ∆CDG 繠∆˹Ҩ ՠ∠C ʹ CD = CG

(8) ѧࡵҠ∆DEG∆EPQ ¤ѹẺ -- (DE = EQ,∠EDG =∠EQP, DG = PQ) ∠DEG =∠PEQ ∠DEG = 6° ∠GEP = 48°
͡ҡ ѧ EG = EP ∆EGP 繠∆˹Ҩ ՠ∠E (= 48°) ʹ ∠EPG (=∠EGP) = 66° ∠DPG = 36°
ԨóҠ∆DGP Ҡ∠DGP = 72° ∠DGP =∠GDP ∆DGP 繠∆˹Ҩ ՠ∠P ʹ DP = GP
ѧࡵҠ∆CDP∆CGP ¤ѹẺ -- (CD = CG, DP = GP, CP = CP) ∠DCP =∠GCP ∠DCP = (∠DCG)/2 = 78° ∠BCP = x = 6° Q.E.D.




Create Date : 15 Ҿѹ 2558
Last Update : 15 Ҿѹ 2558 0:00:00 .
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Fun Geometry Problem with Solution #99



٨ x = 30°
٨ 1



(1) ˹ش P AB CP = BC (= AD) ∆BCP 繠∆˹Ҩ ՠ∠C ʹ ∠BPC =∠CBP ∠BPC = 48° ∠ACP = 30°Р∠APC = 132°

(2) ˹ش Q Ҿз͹ͧش P ҹ AC ∆ACQ∆ACP CQ = CP,∠CAQ =∠CAP = 18°Р∠ACQ =∠ACP = 30°

(3)∵CP = CQ Р∠PCQ = 60° ∆CPQ 繠∆ҹ PQ = CP (= AD),∠CPQ = 60°(∠APQ = 72°) Р∠CQP = 60°
ԨóҠ∆APQ Ҡ∠P = 2(∠A) ըش D AP AD = PQ DQ = PQ(Click ʹԸվ٨)

(4) ѧࡵ CQ = DQ = PQ ش Q 繨شٹҧͧǧՠ∆CDP Ṻ ∠CDP = (∠CQP)/2 x = 30° Q.E.D.

٨ 2



(1) ˹ش P AB CP = BC (= AD) ∆BCP 繠∆˹Ҩ ՠ∠C ʹ ∠BPC =∠CBP ∠BPC = 48° ∠ACP = 30°

(2) AC = a CP = b (AD = b)
˹ش Q ˹ AD AQ = DQ = a ∆ADQ 繠∆˹Ҩ հҹ b дҹСͺʹ a ∠AQD = 24° (Click ʹԸվ٨) ∠DAQ (=∠ADQ) = 78° ∠CAQ = 60°

(3) ∵AC = AQ Р∠CAQ = 60° ∆ACQ 繠∆ҹ CQ = a
ѧࡵ AQ = CQ = DQ ش Q 繨شٹҧͧǧՠ∆ACD Ṻ ∠ACD = (∠AQD)/2 ∠ACD = 12° ∠BDC = x = 30° Q.E.D.




Create Date : 12 Ҿѹ 2558
Last Update : 12 Ҿѹ 2558 23:08:00 .
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Fun Geometry Problem with Solution #98



٨ x = 30°
٨ 1



(1) ԨóҠ∆ABC Ҡ∠ACB = 40°
ԨóҠ∆ABD Ҡ∠ADB = 50° ∆ABD 繠∆˹Ҩ ՠ∠A ʹ AB = AD

(2) ˹ش P BC AP = AB (= AD) ∆ABP 繠∆˹Ҩ ՠ∠A ʹ ∠APB =∠ABP ∠APB = 80° ∠CAP = 40°Р∠APC = 100°

(3)∵AD = AP Р∠DAP = 60° ∆ADP 繠∆ҹ AP = DP Р∠APD = 60°(∠CPD = 40°)
∵ ∠CAP =∠ACP ∆ACP 繠∆˹Ҩ ՠ∠P ʹ AP = CP DP = CP ∆CDP 繠∆˹Ҩ ∠P (= 40°) ʹ ∠DCP (=∠CDP) = 70° ∠ACD = x = 30° Q.E.D.

٨ 2
ѧҡҺ AB = AD Ҩ٨ʹԸǡѹѺ Problem 52




Create Date : 09 Ҿѹ 2558
Last Update : 9 Ҿѹ 2558 0:00:00 .
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Fun Geometry Problem with Solution #97



٨ x = 30°
٨ 1



(1) ˹ش P ☐ABPD 繠☐ѵ AB = AD = BP = DP BC = BP = DP
͡ҡ ѧҠ∠DBP = 45° ∠CBP = 60°

(2)∵BC = BP Р∠CBP = 60° ∆BCP 繠∆ҹ BP = CP Р∠BPC = 60°
ѧࡵ BP = CP = DP ش P 繨شٹҧͧǧՠ∆BCD Ṻ ∠BDC = (∠BPC)/2 x = 30° Q.E.D.

٨ 2



AB (= AD = BC) = a

(1) ˹ش P Шش Q ☐ADQP 繠☐ҡ ըش C 躹 PQ ...
AP = DQ
PQ = AD PQ = a
∠ADQ =∠APQ =∠DQP = 90°

(2)∵AB = AD ∆ABD 繠∆˹Ҩ ՠ∠A (= 90°) ʹ ∠ABD = 45°(∠CBP = 30°) Р∠ADB = 45°(∠BDQ = 45°)
∵ AB = BC ∆ABC 繠∆˹Ҩ ՠ∠B (= 150°) ʹ ∠BAC (=∠ACB) = 15°

(3) Ҡ∆BCP 繠∆ҡ ՠ∠P ҡ Р∠B = 30° CP = BC/2 CP = a/2 CQ = a/2
ѧࡵҠ∆CDQ∆ACP ¤ѹẺ -- (DQ = AP,∠CQD =∠APC, CQ = CP) ∠CDQ =∠CAP ∠CDQ = 15° ∠BDC = x = 30° Q.E.D.




Create Date : 06 Ҿѹ 2558
Last Update : 6 Ҿѹ 2558 0:00:00 .
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Fun Geometry Problem with Solution #96



٨ x = 24°
٨



(1) ∠ACB = 108°
∵ ∠BAC =∠ABC ∆ABC 繠∆˹Ҩ ՠ∠C ʹ AC = BC

(2) ˹ش Q ˹ AB AQ = BQ = AB ∆ABQ 繠∆ҹ ∠ABQ = 60°(∠CBQ = 24°) Р∠AQB = 60°
ѧࡵҠ∆BCQ∆ACQ ¤ѹẺ -- (BC = AC, BQ = AQ, CQ = CQ) ∠BQC =∠AQC = (∠AQB)/2 = 30°

(3) BP ͡ѧش R · BR = BQ BR = AB ∆ABR 繠∆˹Ҩ ՠ∠B (= 12°) ʹ ∠ARB (=∠BAR) = 84°
͡ҡ ѧ ∆BCR∆BCQ ¤ѹẺ -- (BC = BC,∠CBR =∠CBQ, BR = BQ) ∠BRC =∠BQC ∠BRC = 30° ∠CRP =∠CAP ☐APCR öṺǧ ∠ACP =∠ARP ∠ACP = 84° ∠BCP = x = 24° Q.E.D.




Create Date : 03 Ҿѹ 2558
Last Update : 3 Ҿѹ 2558 0:01:10 .
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TIYHz
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