P /jm8c apm( $HjDI4D̜P ;Copyright 1996 - Michael D. Huberty, Ko Hayashi & Chia VangngiH iÌiYPipG  J$JJpͤDt Z LHH$J Y >MCRO OOXOdFOOODTCuOyLHOBC ;@% AXPѐNx *HXPXQ pNxXXCCw ejH AAy4$XH CC 2QMFaphics Apps html Documents M & M Files MATH 5337  ,Squaring A Circle Using Hippias' Quadratrix x x ` Tasks t@oserControl Panels t l havef c]0cFaphics Apps html Documents M & M Files MATH 5337  1`Move point "drag" until it reachs point O. Then the square and circle will have the same area. . els t co`0cqc[c[clc[c`0}hmB4$ 6BXOаNxO:O;HOXOO:XRЀNCCw N N #2  ^PXJPXR0NxXR0XCCw@F Tp m6Tp`Tp j`Tp T BA = gI 6@g e$(B@Distance(B to A) = f @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+ N " 4 c`xnQWѠNBCwD@Cw?p =q N C" 5 c`ynĴO8 0 n anCC߀CB<?Q ?S  OCnCw (" m2OPOp E4O0OOpO OA = gI 6@g e$(B@ radius = B to A) = f @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+ :  radius,2`D//`CnCwCCw?sL6  t={C c20XpXX c|XCnCwC?5%F?5%F )5B0 m37IXBs}ޘCCnC|C~XBIXBs}B(CyC%B  radius = I 6@g e$(B@Length(Segment radius) = f @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+u>yB Ser  p4k Stickies Shut Dwn H$?5%F?5%FCnCwDnCw c4JmID hBCC?5%F?5%F@E   DDnCw ,"_ile  m18 Apps Math Apps etwork Apps User TempAppl  Area p4 = I 6@g e$(B@Area of Circle = dius) = f @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+>C'  PJ IT O CQCM :E  r cX `XX TNDnCwCCw?@ m10_'`Nx E4_'`_'@Nx_ AD = p4 = I 6@g e$(B@Distance(A to D) = us) = f @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+3Y m39mJL ${D:(Area of Circle)}{radius{u:2}} = @g e$(B@,(Area of Circle)/Length(Segment radius)^2 = @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+=LC  re 5 `6D BCCQCM?ޛzo ?1b "drag",@14BCƘCw '  u NxY+p Y$O; O<OY+pO:Y,YCnCnCnBH? ' aaHHO$ICQ(C~,CQC?E'Yile  zcs Apps Math Apps etwork Apps User TempAppl BC~BdC?' s User Temp AppleCD Audo PlayerAutomated Tasks CalCƘB<CƘC߀? ! m5f33ff33ff  A"drag" = f Circle)}{radius{u:2}} = @g e$(B@Distance(A to "drag") = Segment radius)^2 = @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+rw  PCnBH $ m1O3O|0O}pOxO  "drag"O = f Circle)}{radius{u:2}} = @g e$(B@Distance("drag" to O) = Segment radius)^2 = @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+ F` Re m3cscsNI{| {D:L'O}{OA}{!:*}90 = radius{u:2}} = @g e$(B@(Distance(L' to O)/Distance(O to A)*90 = 2 = @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+Z"q  bdQ\B>\BC̘BBF>CnBHCnCw? !I1Re m7cscsNI{| {D:AL'}{BA} = *}90 = radius{u:2}} = @g e$(B@%Distance(A to L')/Distance(B to A) = = 2 = @k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+ PA m4O@`Q WWXOX -L'O{!:*}tan{(:90 {(:{D:L'O}{OA}{!:*}90}} = @g e$(B@FDistance(L' to O)*tan[90 (Distance(L' to O)/Distance(O to A)*90)] = k$($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+1Z"#GRe m8cscsNI{| 5{D:L'O{!:*}tan{(:90 {(:{D:L'O}{OA}{!:*}90}}}{BA} = @g e$(B@Y(Distance(L' to O)*tan[90 (Distance(L' to O)/Distance(O to A)*90)])/Distance(B to A) = ($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+&  ;L>,2`D//CƘC.QE' %  tCnCwCƘC.QE?( 6J ('̙ ?JY0 Nx (0YY Nx 08<Y>((" #&%'(@  m15 @S@p@HH48  L"drag" = }tan{(:90 {(:{D:L'O}{OA}{!:*}90}}}{BA} = @g e$(B@Distance(L to "drag") = 0 (Distance(L' to O)/Distance(O to A)*90)])/Distance(B to A) = ($  0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+(  ab[arctan[-abs[sqrt[ln[log[round[nCƘCwCƘC.QE?(g:j m27ssss uuu u     ={D:AD{!:*}AD}{2{!:*}{(:Q'L' (drag)}{!:*}{(:Q'L' (drag)}} = @g e$(B@^Distance(A to D)*Distance(A to D)/(2*Distance(Q' to L' (drag))*Distance(Q' to L' (drag))) =   0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+ +[gZj m19ssss uuu u      {D:AD}{Q'L' (drag)} = Q'L' (drag)}{!:*}{(:Q'L' (drag)}} = @g e$(B@AD/("drag"L) = )*Distance(A to D)/(2*Distance(Q' to L' (drag))*Distance(Q' to L' (drag))) =   0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55++H3O| m16<ff33ff ({D:AD{!:*}AD}{2{!:*}{(:Q'L' (drag)}} = }{(:Q'L' (drag)}} = @g e$(B@CDistance(A to D)*Distance(A to D)/(2*Distance(Q' to L' (drag))) = tance(Q' to L' (drag))) =   0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+ +  c5dRHHO BCB]v?5%F?5%F,MRH  T$HHBCwQF 0iMnR &T$HHBW w 1-ECRnsla  atlly and by distance ^4 verticaly^1.W^0. Let ^2 = Image of ^33 translatbBCBW w?2hLCR  arXHGW g VxVXUUBW wBC?2au' aoGx  VxgdUWtdCZ*AVG?2in XHI  4CQ:  5hu Hel  avr ObliqueI Helvetica ObliqeI New Century Schlbk ItalicImCQ: BW w?26 ^33 c75 about center ^44^1.H^0. Let ^ = Image of ^33 dilated by scale factor BW wC?5%F?5%F26kOA  p6W,UUUUdXHHf?CQ: BW wBCCQCM62hCH  asH$GXCQCMCQ: ?6hLn  aqaphics Apps Math Apps Network Apps User TemppCQ: BW w?26MR0000 AB111`8R\NHB)Ë+ 38LCR  auB)Ë+BC?<MRTime Oe(U< )7hBA =#A|ic c8City Medium ItalicCODABAR1 CODE3OF9X1CourierDorchestSBAC?5%F?5%F>iUnZTim Fe( `)7( 7?hTnR  awx$Rx`8RHNUBW w(?@2p^  c9G|  CCCwu?5%F?5%FAej! ACI X G AQ@Cy Bo&_  c10etwork Apps User TempAppleCD Audio PlayerAutomateaAQ@CyCwu?5%F?5%FCdj  axI|gg ag,dXAQ@CyCC?C'#H az$HXACk8hACW ?EC'f ayDHicki(XC*CkEC|C]v?Emr! AEi a4$XA춠B FDns ADZ b,Zh bZfVgCUBQ GBoh  p7UUDD""?""A춠BCUBQCCAQ@CyHIClj!33  bcff33̙ffA춠BAQ@Cy?CHls  bbZhg8ihdZhHCUBQA춠B?HImj  baXZ$CUBQCC?I+b m29llll*lll Area ACAAADAE = !:*}{(:Q'L' (drag)}} = }{(:Q'L' (drag)}} = @g e$(B@Area of Square = Distance(A to D)/(2*Distance(Q' to L' (drag))) = tance(Q' to L' (drag))) =   0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+JYae m30mHHWKXXX ZZZ Z hhh hj  {V:(Area of Square)} = Q'L' (drag)}} = }{(:Q'L' (drag)}} = @g e$(B@side = Square = Distance(A to D)/(2*Distance(Q' to L' (drag))) = tance(Q' to L' (drag))) =   0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+5N_ m38mHRPROPOP ${D:(Area of Square)}{radius{u:2}} = } = }{(:Q'L' (drag)}} = @g e$(B@,(Area of Square)/Length(Segment radius)^2 = e(Q' to L' (drag))) = tance(Q' to L' (drag))) =   0f0(((@/|hI 6efDHD ff 3>!=tzhT3 f g |gmghThTg"He @[2L6 g`5n P ng$ g D($g55+N