2pkrcapmU  2N@^8/)p#v ?fTo try your angle experiments: 1. Set your script tools folder (Preferences in the Display menu, then More) to the Poincar disk folder. 2. Use the Hyperbolic Segment tool to draw the triangle. 3. Use the Hyperbolic Angle tool to measure angles.cX\uaz 2N@^8/)p#Bv  F FTObCB ch 2N@^8/)p#Av  F FTObCCf  2N@^8/)p# P-Disk Radius[[ F Fp[?C؀C?MR 2N@^8/)p# P-Disk Center[[ F Fp[?CC   2N@^8/)p#Cv  F FTObCtC  l & 2N@^8/)p# !Fj )LMoThe Poincar disk is a model for hyperbolic geometry. In the model, a straight line through two points is defined as an arc passing through the two points and perpendicular to the circle. Play with the hyperbolic triangle to discover things about angles. How large can the sum of the angles of a hyperbolic triangle be? How small? Bill Finzer, 3/95, with much help from Mike AlexanderM`5 2N@^8/)p#Hide ExperimentK 0K@ @K@  k#KK@EB ey Cha0C9 2N@^8/)p#Show Experiment@qkKk%p k)BK@ [bz  2N@^8/)p#oCCfCB?L' tems  2N@^8/)p#jsAutomated TasksBil Fs Remote ChooserControaCC CCf? 2N@^8/)p#Diskv  F FTObCC B O?YN }L# $  2N@^8/)p#s0 H`CJUUUUCC CB?[z  2N@^8/)p#xCBCtC ?h  2N@^8/)p#abDH$uCCfCtC ?'# $ 2N@^8/)p#azTONOUUUU,CC CB?9' 2N@^8/)p#bu xCC CtC ?'  2N@^8/)p#cpCC CCf?lq0 2N@^8/)p#FCC" 'por 2N@^8/)p#kCC[C@Cq?  `' 2N@^8/)p#t$CBC@B? r4w9 2N@^8/)p#MCB +0 2N@^8/)p#QOvxC@C7 7< 2N@^8 /)p#APMl<C'Bd   tems 2N@^8 /)p#AQAutomated TasksBil Fs Remote ChooserControaCC`~   2N@^8 /)p#BL  hh,rCaC+0  h, 2N@^8 /)p#BMqr0CtC  gl 2N@^8 /)p#CH<CPCwʹ  3%8 2N@^8 /)p#CICLB/  C' 2N@^8/)p#qXt''XhCCCC+? (- 2N@^8 /)p#ECCt@  t  2N@^8 /)p#K h PC-CT  :'  2N@^8/)p#zCC5CBH?'!  2N@^8/)p#acuC~CpCB??I 2N@^8/)p#AQAutomated TasksBil Fs Remote ChooserControaCCE '#  2N@^8/)p#bgC'BdCC`~??I 2N@^8/)p#BMqr0CG 9' 2N@^8/)p#cbCaC+0CtC??I 2N@^8/)p#CIC-zC '  2N@^8/)p#cwCPCwʹCLB/?'R  2N@^8/)p#lCCt@CC ?L  2N@^8/)p#u qrC5chC-CTCC ? j' 2N@^8/)p#ba UC'BdCCE?#'  2N@^8/)p#bcCCECB?# -' 2N@^8/)p#beCCECC`~?#ݍ%' 2N@^8/)p#bvCaC+0CG?%'& 2N@^8/)p#bxCGCtC ?%Q' 2N@^8/)p#bzCGCtC?%' 2N@^8/)p#cqCPCwʹC-zC?'' 2N@^8/)p#csC-zCCCf?':' 2N@^8/)p#cuC-zCCLB/?'' 2N@^8/)p#mCv) CbCC?)~' 2N@^8/)p#v$C,CO" 2N@^8 /)p#CJCCXKe 1 mr 2N@^8/)p#InvertedCWOC 4 2N@^8/)p#LCN 5 ݎ'&" 2N@^8/)p#bbINv ?Jw VC˫B7CC`~?6.' 2N@^8/)p#bwCfC5CtC?7'X 2N@^8/)p#crCCXKeCLB/?8br  2N@^8/)p#nCWOCCCf?9ta  2N@^8/)p#wCNCB?:dwi| 2N@^8/)p#ASC5B ,; 2N@^8/)p#BOCxC+ /<38 2N@^8/)p#CKCCME 2=hmw  2N@^8/)p#G'ZCCy >). 2N@^8/)p#NCGB+J ?' 2N@^8/)p#bdBG~  WC'BdC5B?@ S' 2N@^8/)p#byCaC+0CxC+?A' 2N@^8/)p#ctCPCwʹCCME?B ' 2N@^8/)p#pY''YC C~lCpCtA?>C݅ ' 2N@^8/)p#yCݢBC`;D??D#( 2N@^8/)p#ATChCj0 -Edi 2N@^8/)p#BPChB'M 0Fqv 2N@^8/)p#CLCGBM 3GPU 2N@^8/)p#HWtt''W@DdC@. H 2N@^8/)p#OCj"HF !I  2N@^8/)p#RCXC "H' 2N@^8/)p#bfM R XC˫B7ChCj0?J6s' 2N@^8/)p#caCfC5ChB'M?K7#' 2N@^8/)p#cvCCXKeCGBM?L8CUtems  2N@^8/)p#rsAutomated TasksBil Fs Remote ChooserControaDdC@.CC4AB?MbCu| 2N@^8/)p#c2V\DdC@.Cp ?5%F?5%FM9 2N@^8/)p#c3Cj"HFC{U?5%F?5%FN:'   2N@^8/)p#aaCj"HFCIp۱?NBq 2N@^8/)p#c4CXCC5?5%F?5%FO9 !  2N@^8/)p#adCXCC0C]k?O 2N@^8/)p#AUC  $P 2N@^8/)p#BQCJ>C*& &Q mr 2N@^8/)p#CMCWOC (Rbg 2N@^8 /)p#IC C  ST{7< 2N@^8 /)p#PC@BQ0 VU7< 2N@^8 /)p#SCBcC( XWL  2N@^8/)p#bh5: YCC C ?Ybd  2N@^8/)p#bi  c6CCfC ?Y'  2N@^8/)p#bkJ 3X3+0$CCfC ?Y  2N@^8/)p#boCtC C ?Y 7' 2N@^8/)p#bq?!D&  AHCtC C ?YR  2N@^8/)p#ccCC CJ>C*&?Z[z  2N@^8/)p#cdCBCJ>C*&?Zf ' 2N@^8/)p#cfCBCJ>C*&?Zh  2N@^8/)p#cjCCfCJ>C*&?Z' 2N@^8/)p#clCCfCJ>C*&?ZLr  2N@^8/)p#cxCC CWOC?[r  2N@^8/)p#cyCtC CWOC?[0'$ 2N@^8/)p#daCtC CWOC?[[lz  2N@^8/)p#deCBCWOC?['Y 2N@^8/)p#dgCBCWOC?[^bwtems 2N@^8 / )p#a1Automated TasksBil Fs Remote ChooserControaDcC@.Cp }m[CBCCfAHA<bC?\^w 2N@^8 / )p#a2Cj HGC{U>?s;OCtC CBA|Ar?]e 2N@^8 / )p#a3CXCC-?"E#CCfCtC AxUhAcBq?^@E 2N@^8/)p#AVCGBo _ 2N@^8/)p#c11C C@?5%F?5%FY`i'D 2N@^8/)p#bj)  c7Cwds`DRVB0?`Y 2N@^8/)p#c12C C ?5%F?5%FYb' 2N@^8/)p#bpC·d0DVBf?bY  2N@^8/)p#BRCOC  d9| 2N@^8/)p#c15CJ>C*&C1X?5%F?5%FZeX'  2N@^8/)p#ceC^C]C5BC&P?eZ= 2N@^8/)p#c16CJ>C*&C5O?5%F?5%FZg*' 2N@^8/)p#ckCCTCyBg&P?gZ]b 2N@^8/)p#CNCkCM iY# 2N@^8/)p#c19CWOCC4T?5%F?5%F[j&' 2N@^8/)p#czCbCOCLCA8?j[^ 2N@^8/)p#c20CWOCC/]?5%F?5%F[l' 2N@^8/)p#dfC"CCC8?l[, 2N@^8/)p#c9 J<8J<? DCGBoB?5%F?5%FqY 2N@^8 /)p#AXC_A sr_d 2N@^8 /)p#AYt|LKPD/B sr;!@ 2N@^8 /)p#BEC`bv ut]b " 2N@^8 /)p#BF  aDݨCQ utHP 2N@^8/)p#c13COC B?5%F?5%FvZ27 2N@^8 /)p#BTCsiC xw 2N@^8 /)p#BUC![ xw i%n 2N@^8 /)p#CABC zy', 2N@^8 /)p#CBC[ zy 2N@^8/)p#c17CkCMB?5%F?5%F{[ 2N@^8 /)p#CPClC| }| 2N@^8 /)p#CQCjC }|! 2N@^8 /)p#CWDvoC@ ~ 2N@^8 /)p#CXC?~Cl ~!0&5L 2N@^8/)p#AWh  c5CB   2N@^8/)p#BSCqJ\CF/  2N@^8/)p#COCkCWg U,q 2N@^8/)p#c10&+  AAC C"[?5%F?5%FY= 2N@^8/)p#c14CJ>C*&B 0?5%F?5%FZ&W 2N@^8/)p#c18CWOCB?5%F?5%F[ 2N@^8 /)p#AZCLC 2 aaf 2N@^8 /)p#BAx}   ADD a8= 2N@^8 /)p#BBCBDI+B5z-! 'Cb™OL CC“ s7a<f 2N@^8 /)p#BC`DfBd sPEUJ 2N@^8 /)p#BGCB]i cSX 2N@^8 /)p#BHD*  c8DMǬ cortw 2N@^8 /)p#BI   AIC et u}&+ 2N@^8 /)p#BJ   AJD UB uty 2N@^8 /)p#BVC^B f 2N@^8 /)p#BWC\CpX f 2N@^8 /)p#BXC[CO x=B 2N@^8 /)p#BYC8Bz x  2N@^8 /)p#CCCC. h^c 2N@^8 /)p#CDC 1Bs| h 2N@^8 /)p#CEC#NuCF zFK 2N@^8 /)p#CFCq/EB> z;@ 2N@^8 /)p#CRCCW kAF 2N@^8 /)p#CSC*CT k 2N@^8 /)p#CTC`C[ }=7B< 2N@^8 /)p#CUC?Ci/ }pu 2N@^8 /)p#CYClCDC mTiYn 2N@^8 /)p#CZC2C 8 m 2N@^8 /)p#DACCP $ ) 2N@^8 /)p#DBCCs ,^' 2N@^8/)p#blCLC 2D/B?' 2N@^8/)p#brN"'  CB]iDݨCQ?+'N 2N@^8/)p#cgC^BC![?'y 2N@^8/)p#cmCC.C[?S' 2N@^8/)p#dbCCWCjC?'m' 2N@^8/)p#dhClCDCC?~Cl?)' 2N@^8/)p#bmC]BD>B ?J|' 2N@^8/)p#bnCAjC!PC|¿+B?\' 2N@^8/)p#bs  avCjBzD-C>C?' 2N@^8/)p#btNdW\   CgKO~dDRž.?' 2N@^8/)p#chCqC iC$Xq.?' 2N@^8/)p#ciC,CC"C,?'O 2N@^8/)p#cnCaC(xACjnW?q' 2N@^8/)p#coCCC2 B˸?' 2N@^8/)p#dcC98CfCFCt_?'Z 2N@^8/)p#ddCXCmCuhC#?Q' 2N@^8/)p#diCCiiCLvC?' 2N@^8/)p#djD9CpVbC"Cv)?RWC 2N@^8/)p#BDCҤBvX a-x2} 2N@^8/)p#BK   ALC6!B=6 c~ 2N@^8/)p#BZChB f 2N@^8/)p#CGCdUC'% hY^ 2N@^8/)p#CVCJCw kns 2N@^8/)p#DCCCnӟ m 2N@^8/)p#m2  aw Angle = aa8Ξk!0z)Š8Ξk#J@>;HH! Angle (ABC) = lK@kb(k D/!Uk @4@6@6@6$$!VVN$ k(za  N$*A*A*A8Ξ#J*'<'k#*>k#`k#J `f@ k#Jk!Ij''<*k0 y :Bi8Τk"0z8Τk%K K@Kk92#@h@8Ξk#0z)Š85P  K<@K k * 2N@^8/)p#m3 Angle = aa8Ξk!0z)Š8Ξk#J@>;HH! Angle (BCA) = lK@kb(k D/!Uk @4@6@6@6$$!VVN$ k(za  N$*A*A*A8Ξ#J*'<'k#*>k#`k#J `f@ k#Jk!Ij''<*k0 y :Bi8Τk"0z8Τk%K K@Kk92#@h@8Ξk#0z)Š85P  K<@K k +8 2N@^8/)p#m4 Angle = aa8Ξk!0z)Š8Ξk#J@>;HH! Angle (CAB) = lK@kb(k D/!Uk @4@6@6@6$$!VVN$ k(za  N$*A*A*A8Ξ#J*'<'k#*>k#`k#J `f@ k#Jk!Ij''<*k0 y :Bi8Τk"0z8Τk%K K@Kk92#@h@8Ξk#0z)Š85P  K<@K k ?L 2N@^8/)p#m5d TL  0(Angle (ABC)) + (Angle (BCA)) + (Angle (CAB)) = aa8Ξk!0z)Š8Ξk#J@>;HH! Sum of angles of the triangle = k D/!Uk @4@6@6@6$$!VVN$ k(za  N$*A*A*A8Ξ#J*'<'k#*>k#`k#J `f@ k#Jk!Ij''<*k0 y :Bi8Τk"0z8Τk%K K@Kk92#@h@8Ξk#0z)Š85P  K<@K k /6  2N@^8/)p#Hide Explanation@qkKk%  pon;  2N@^8/)p#Show Explanation@qkKk% P elspon# 2N@^8/)p#Resetk%@qkKk% # 2N@^8/)p# Try your own!@qkKk%*