We must continue with the Arbelos!

by Klaus-Anton, Dec 5, 2018, 6:07 PM

As you know - as gentle reader of my blog - through my idea to make use of the arbelos the Golden Ratio further could be found in the classical cardioid. At that time i was too weak to find it alone. I had calculation errors with the extrempoints. So i had asked the famous swissian mathematician Hans Walser.

Let us fetch the two drawings of the arbelos by marris, Figure 1 and Figure 2:

$[asy] /* Figure 0034: fig_rc04_160308_arcs.asy http://asy.marris.fr/asymptote/Cercles/index.html */ size(7cm,0); // On définit... pair O=(0,0),A1=(-1,0),A2=(3,0); real r=4, r1=3, r2=1; path chemin=buildcycle(arc(O,r,0,180),arc(A1,r1,0,180),arc(A2,r2,0,180)); pen stylo=1bp+blue, pinceau=lightgrey; // ... puis on dessine. filldraw(chemin,pinceau,stylo); draw(A1+r1*W--A2+r2*E,dashed+red); dot(O);draw(Label("$r$"),O--r*dir(60),Arrow); dot(A1);draw(Label("$r_1$"),A1--A1+r1*dir(90),Arrow); dot(A2);draw(Label("$r_2$"),A2--A2+r2*dir(120),Arrow); // une bordure blanche de 3mm autour de l'image shipout(bbox(3mm,white)); [/asy]$ $[asy]label("Figure 1");[/asy]$

$[asy] /* http://asy.marris.fr/asymptote/Cercles/index.html Figure 0035: fig_rc05_160308_arcs.asy */ size(7cm,0); real h=5; pair O=(0,0),A=(h,0),B=(-h,0); // M(r=h,theta=70) point du cercle de diamètre [AB] pair M=h*dir(70); // H, projeté orthogonal de M sur l'axe (AB) des abscisses pair H=(M.x,0); // On définit le centre des demi-cercles. pair A1=(A+H)/2, B1=(B+H)/2; // On définit les stylos qui seront utilisés. pen stylo=1bp+blue, pinceau=lightgrey; // On définit les arcs de cercles. path dc=arc(O,A,B),dca=arc(A1,A,H),dcb=arc(B1,H,B); // ... et on trace le tout : filldraw(buildcycle(dc,dca,dcb),pinceau,stylo); dot(O); dot("$M$",M,dir(O--M)); draw(A--B^^O--M--H^^A--M--B,dashed); // en ajoutant deux points d'intersection... dot("$N$",intersectionpoint(reverse(dca),M--A),N); dot("$P$",intersectionpoint(dcb,M--B),N); // ... une bordure blanche de 3mm autour de l'image shipout(bbox(3mm,white)); [/asy]$ $[asy]label("Figure 2");[/asy]$

$[asy] size(7cm,0); real h=5; pair O=(0,0),A=(h,0),B=(-h,0); // M(r=h,theta=76) point du cercle de diamètre [AB] pair M=h*dir(76); // H, projeté orthogonal de M sur l'axe (AB) des abscisses pair H=(M.x,0); // On définit le centre des demi-cercles. pair A1=(A+H)/2, B1=(B+H)/2; // On définit les stylos qui seront utilisés. pen stylo=1bp+blue, pinceau=lightgrey; // On définit les arcs de cercles. path dc=Arc(O,A,B),dca=Arc(A1,A,H),dcb=Arc(B1,H,B); // ... et on trace le tout : filldraw(buildcycle(dc,dca,dcb),pinceau,stylo); draw(A--B^^O--M--H^^A--M--B,dashed); real gr=(sqrt(5)-1)/2; real GR=(sqrt(5)+1)/2; draw("$38^\circ$",Arc(B,GR,0,76/2),PenMargins); draw("$104^\circ$",Arc(O,GR^1*gr,76,180),.25RightSide,red,PenMargins); pair P=intersectionpoint(dcb,M--B); draw(O--(O.x,h),dotted); draw(P--(P.x,0),blue+dotted,PenMargins); draw(Label("$76^\circ$",Relative(.4)),Arc(O,1,0,76),.1*LeftSide,fontsize(5)); draw(Label("$52^\circ$",Relative(.675)),Arc(A,1,180-52,180),.1*LeftSide,fontsize(5)); draw(Label("$38^\circ$",Relative(.45)),Arc(M,1,-90,-90+38),.3*RightSide,fontsize(5)); draw(Label("$38^\circ$",Relative(.6)),Arc(M,1,-90-14-38,-90-14),.8*RightSide,fontsize(5)); draw(M--P,linewidth(2)+red); draw(B--P,linewidth(2)+blue); //draw(Arc(B,arclength(B--P),0,90),blue); // en ajoutant deux points d'intersection... pair Q=intersectionpoint(reverse(dca),M--A); import geometry; line linePQ=line(P,Q); draw(linePQ,lightblue+white); draw(circle(M,P,Q),lightblue+white); pair R=intersectionpoint(P--Q,M--O); //pair S=intersectionpoint(P--Q,M--(M.x,0))); dot("$R$",R,2*dir(R-- midpoint(P--M)),fontsize(5)); /* label("$R$", R//, dir(NW) // //dir(R-- midpoint(P--M)) ); */ pair third_gold_candidat=relpoint(Q--P,gr); draw(P--third_gold_candidat,red+linewidth(2)); draw(Q--third_gold_candidat,blue+linewidth(2)); dot(third_gold_candidat, FillDraw(yellow)); dot("$Q$",Q,dir(-90-45/1)); dot("$P$",P,dir(90-14)); dot("$M$",M,dir(O--M),UnFill); dot((M.x,0),blue,UnFill); dot("$(P.x,0)$",(P.x,0),S,blue+fontsize(5),UnFill); dot("$O$",origin,S,red,UnFill); dot("$A$",A,SE,UnFill); dot("$B$",B,SW,UnFill); pair first_gold_candidat=relpoint(B--M,gr); pair second_gold_candidat=relpoint(Arc(O, h,0,180),gr); //draw(second_gold_candidat--(second_gold_candidat.x,0),dotted,PenMargins); dot(first_gold_candidat, FillDraw(yellow)); //dot(second_gold_candidat, FillDraw(yellow)); pair help_point[]; help_point[0]=relpoint(B--A, 0.615661475326); // sin(38) help_point[1]=relpoint(B--A, 0.788010753607); // cos(38) help_point[2]=relpoint(B--(M.x,0), 0.788010753607); help_point[3]=(O.x,h); help_point[4]=midpoint(P--M); help_point[5]=(M.x,h*0.615661475326); help_point[6]=midpoint(B--(M.x,0)); help_point[7]=midpoint(A--(M.x,0));// Mittelpunkt kleiner Kreis help_point[8]=relpoint(help_point[3]--O,.5);//knapp auf blauem Kreis label("$O^\prime$", help_point[3],N,red); label("$(M.x,0)$",(M.x,0),S,blue+fontsize(5)); /* dot(help_point[0] // not on the intersection of the small blue circles ^^help_point[1] ^^help_point[2] ^^help_point[3] ^^help_point[4] ^^help_point[5] ^^help_point[6]//Mittelpunkt mittlerer Kreis ^^help_point[7]//Mittelpunkt kleiner Kreis ^^help_point[8] //^^help_point[] ,UnFill); */ dot(help_point[6] ^^help_point[7] ,blue,UnFill); dot(help_point[3],red,UnFill);//O^\prime //dot(relpoint(help_point[3]--O,.5));//knapp draw(intersectionpoint(P--Q, M--O));//knapp // ... une bordure blanche de 3mm autour de l'image shipout(bbox(3mm,white)); [/asy]$ $[asy]label("Figure 3");[/asy]$

Figure 3 is a modification of Figure 2. The point M is defined as $\angle(AOM)=76^\circ$ . So it follows that $\angle(MOB)=(90+14)^\circ=104^\circ$ . We have: $38+104=142$ and so we get $180-142=38$ . This says us: $\angle(OMB)=\angle(OBM)38^\circ$ .

At $M$ we have a right angle: $\angle(BMA)=90^\circ$ . As the angle at $B$ is known to us to be $38^\circ$ , we have that $\angle(AMB)=52^\circ$ . $(180-(90+38)=52)$ . We have further $\angle(OM(M.x,0))=14^\circ$ , because of: $(180-(90+76)=14)$ .

The $\sin(38^\circ)=0.615661475326\ldots\approx\Phi=0.618\ldots$ .
The $\cos(38^\circ)=0.788010753607\ldots$

The in red/blue marqued linesegments fullfill exactly the Golden Ratio. (Therefore the cut is marqued in yellow.)
But these yellow points are not identical and congruent with $P$ and $R$ , if $M$ is defined to have the direction $\theta=76^\circ$ . It must be some what bigger: $\theta\approx76.324$ . But at what value in degree exactly? i ask me.

Here are some remarques about the Arbelos.

Here are some further remarques especially about the Golden Arbelos.

$[asy] size(8cm,10cm); unitsize(2.3cm); pair A,B,C; real GR=(sqrt(5)+1)/2; A=(0,0); B=(2,0); C=(1,0); path arc_1=Arc(C,1,0,180) ,arc_2=Arc(A,GR,0,45) ,arc_3=Arc(C,1,0,72) ; pair M=intersectionpoint(arc_1,arc_2); path arc_4=Arc(A,(GR,0),M); draw(arc_4,magenta); draw(arc_1); //draw(arc_4); draw("$72^\circ$",arc_3,magenta+linewidth(2)); draw(C--M,magenta+linewidth(2)); draw(A--M--B); draw(M--(M.x,0),dotted); draw(C--(GR,0),blue+linewidth(2)); draw((GR,0)--B,red+linewidth(2)); draw(A--C); label("$A$",A,SW); label("$B$",B,SE); label("$M$",M,NNE); label("$C$",C,S); label("$D$",(M.x,0),S); label("$E$",(GR,0),S); dot(A^^B^^C^^M,UnFill); dot((GR,0), FillDraw(yellow)); usepackage("amsmath"); label( "$CE/CB=\frac{\sqrt5-1}{2}=\,\mathpunct{.}618\ldots$" // "$CE/CB=\dfrac{\sqrt5-1}{2}=\,.618\ldots$" ??!? ,truepoint(2S),S); label("1",C--A,fontsize(8)); shipout(bbox(3.5mm,3.5mm,FillDraw( yellow+.8blue ,orange+.5blue +GR*mm+miterjoin) )); [/asy]$ $[asy]label("Figure 4");[/asy]$
Figure 4 is also not the exact solution for the problem i am trying to solve. The angle $\theta$ must be greater than $72^\circ$ . It is also greater than $76^\circ$ . It is about $\theta\approx76.324^\circ$ .

$[asy] import geometry; usepackage("amsmath"); size(8cm,10cm); unitsize(2.3cm); point A,B,C; real GR=(sqrt(5)+1)/2; real gr=(sqrt(5)-1)/2; A=(0,0); B=(2,0); C=(1,0); path arc_1=arc(C,1,0,180) ; draw(arc_1); /* pair M=intersectionpoint(arc_1,arc_2); path arc_4=Arc(A,(GR,0),M); draw(arc_4,magenta); draw(arc_1); //draw(arc_4); draw("$72^\circ$",arc_3,magenta+linewidth(2)); draw(C--M,magenta+linewidth(2)); */ //draw(A--M--B); /* draw(M--(M.x,0),dotted); draw(C--(GR,0),blue+linewidth(2)); draw((GR,0)--B,red+linewidth(2)); */ point D=C+gr^3; line perp=perpendicular(D, line(A,B)); //draw(perp); point Dprime=(D.x,1); //dot(Dprime); point M=intersectionpoint(D--Dprime,arc_1); draw(A--M--B^^M--D^^C--B^^M--C); /* point M=intersectionpoint(perp, arc_1); no matching function 'intersectionpoint(, path) */ //point M=intersectionpoint(arc_1,perp); ambigous draw(A--C); //draw(Arc((pair)C,(pair)B,(pair)M),magenta); draw(rotate(degrees(B-M))*format("$%f6\ldots^\circ$",degrees(M-C)),Arc((pair)C,(pair)B,(pair)M),magenta); draw(B--C--M,magenta); draw("$1$",(A.x,-.3)--(C.x,-.3),LeftSide,Bars); dot(M,UnFill); dot(D,UnFill); label("$A$",A,SW); label("$B$",B,SE); label("$M$",M,dir(midpoint(C--D)--M)); label("$C$",C,S); label("$D$",D,SE); //label(slant(-.28)*"$\varphi$", (1,0), fontsize(25)+yellow); label(slant(-.28)*"$\varphi$",D--C,magenta); label("$\phantom{\varphi\,}^3$",D--C,magenta); //label("$E$",(GR,0),S); dot(A ^^B ^^C //^^M ,UnFill); //dot((GR,0), FillDraw(yellow)); /* label( "$CE/CB=\frac{\sqrt5-1}{2}=\,\mathpunct{.}618\ldots$" // "$CE/CB=\dfrac{\sqrt5-1}{2}=\,.618\ldots$" ??!? ,truepoint(2S),S); */ //label("1",C--A,fontsize(8)); point m1=midpoint(A--D); point m2=midpoint(B--D); draw(Arc(m1,arclength(A--m1),0,180),blue); draw(Arc(m2,arclength(B--m2),0,180),blue); draw(M--relpoint(M--A,gr^2),red+linewidth(2)); draw(A--relpoint(M--A,gr^2),blue+linewidth(2)); point goldpoint_1=relpoint(M--A,gr^2); dot(m1^^m2,blue,UnFill); dot(goldpoint_1, FillDraw(yellow)); dot(A^^D^^B^^M, UnFill); shipout(bbox(3.5mm,3.5mm,FillDraw( yellow+.8blue ,orange+.5blue +GR*mm+miterjoin) ));[/asy]$ $[asy]label("Figure 5, phi\ $=\frac{\sqrt5-1}{2}$");[/asy]$
$[asy] import geometry; usepackage("amsmath"); size(8cm,10cm); unitsize(3.5cm); point A,B,C; real GR=(sqrt(5)+1)/2; real gr=(sqrt(5)-1)/2; A=(0,0); B=(2,0); C=(1,0); path arc_1=arc(C,1,0,180) ; draw(arc_1); /* pair M=intersectionpoint(arc_1,arc_2); path arc_4=Arc(A,(GR,0),M); draw(arc_4,magenta); draw(arc_1); //draw(arc_4); draw("$72^\circ$",arc_3,magenta+linewidth(2)); draw(C--M,magenta+linewidth(2)); */ //draw(A--M--B); /* draw(M--(M.x,0),dotted); draw(C--(GR,0),blue+linewidth(2)); draw((GR,0)--B,red+linewidth(2)); */ point D=C+gr^3; line perp=perpendicular(D, line(A,B)); //draw(perp); point Dprime=(D.x,1); //dot(Dprime); point M=intersectionpoint(D--Dprime,arc_1); draw(A--M--B^^M--D^^C--B^^M--C); /* point M=intersectionpoint(perp, arc_1); no matching function 'intersectionpoint(, path) */ //point M=intersectionpoint(arc_1,perp); ambigous draw(A--C); //draw(Arc((pair)C,(pair)B,(pair)M),magenta); dot(M,UnFill); dot(D,UnFill); label("$A$",A,SW); label("$B$",B,SE); label("$M$",M,dir(midpoint(C--D)--M)); label("$C$",C,S); label("$D$",D,S); //label(slant(-.28)*"$\varphi$", (1,0), fontsize(25)+yellow); label(slant(-.28)*"$\varphi$",D--C,magenta); label("$\phantom{\varphi\,}^3$",D--C,magenta); //label("$E$",(GR,0),S); dot(A ^^B ^^C //^^M ,UnFill); //dot((GR,0), FillDraw(yellow)); /* label( "$CE/CB=\frac{\sqrt5-1}{2}=\,\mathpunct{.}618\ldots$" // "$CE/CB=\dfrac{\sqrt5-1}{2}=\,.618\ldots$" ??!? ,truepoint(2S),S); */ //label("1",C--A,fontsize(8)); point m1=midpoint(A--D); point m2=midpoint(B--D); point goldpoint_1=relpoint(M--A,gr^2); point P=goldpoint_1; point Q=intersectionpoint(M--B, Arc(m2,arclength(B--m2),1,180)); point goldpoint_2=relpoint(Q--P,gr); point R=goldpoint_2; point myS=midpoint(P--Q); point T=midpoint(A--M); draw(Arc(m1,arclength(A--m1),0,180),blue); draw(Arc(m2,arclength(B--m2),0,180),blue); draw(M--relpoint(M--A,gr^2),red+linewidth(2)); draw(A--relpoint(M--A,gr^2),blue+linewidth(2)); draw(Q--goldpoint_2,blue+linewidth(2)); draw(P--goldpoint_2,red+linewidth(2)); draw(Circle(myS, arclength(M--myS)),lightgrey); //draw(Circle(T,.5*arclength(M--A)),lightgrey); draw(rotate(degrees(B-M))*format("$%f6\ldots^\circ$",degrees(M-C)),Arc((pair)C,(pair)B,(pair)M),magenta); draw(B--C--M,magenta); draw("$1$",(A.x,-.3)--(C.x,-.3),LeftSide,Bars); //label("$P$",P,I*dir(A--M)); label("$P$",P,dir(C--P)); label("$R$",R,3*dir(R--relpoint(M--P,.55)),fontsize(5)); label("$S$",myS,2*dir(myS--relpoint(M--B,.3)),fontsize(5)); dot(m1^^m2,blue,UnFill); dot(goldpoint_1, FillDraw(yellow)); dot(goldpoint_2, FillDraw(yellow)); dot(A^^C^^D^^B^^M^^myS, UnFill); dot(Q,UnFill); dot(T,UnFill); label("$T$",T,1.5*I*dir(A--M),fontsize(5)); label("$Q$",Q,dir(Q--midpoint(D--m2))); //dot(relpoint(M--C,.6)); label( "$AP/PM=QR/RP=\frac{\sqrt5-1}{2}=\,0\mathpunct{.}618\ldots$" // "$CE/CB=\dfrac{\sqrt5-1}{2}=\,.618\ldots$" ??!? ,truepoint(2.5S),S); shipout(bbox(3.5mm,3.5mm,FillDraw( yellow+.8blue ,orange+.5blue +GR*mm+miterjoin) ));[/asy]$ $[asy]label("Figure 6");[/asy]$

$[asy] import geometry; usepackage("amsmath"); size(8cm,10cm); unitsize(3.5cm); point A,B,C; real GR=(sqrt(5)+1)/2; real gr=(sqrt(5)-1)/2; A=(0,0); B=(2,0); C=(1,0); path arc_1=arc(C,1,0,180) ; draw(arc_1); /* pair M=intersectionpoint(arc_1,arc_2); path arc_4=Arc(A,(GR,0),M); draw(arc_4,magenta); draw(arc_1); //draw(arc_4); draw("$72^\circ$",arc_3,magenta+linewidth(2)); draw(C--M,magenta+linewidth(2)); */ //draw(A--M--B); /* draw(M--(M.x,0),dotted); draw(C--(GR,0),blue+linewidth(2)); draw((GR,0)--B,red+linewidth(2)); */ point D=C+gr^3; line perp=perpendicular(D, line(A,B)); //draw(perp); point Dprime=(D.x,1); //dot(Dprime); point M=intersectionpoint(D--Dprime,arc_1); draw(A--M--B^^M--D^^C--B^^M--C); /* point M=intersectionpoint(perp, arc_1); no matching function 'intersectionpoint(, path) */ //point M=intersectionpoint(arc_1,perp); ambigous draw(A--C); //draw(Arc((pair)C,(pair)B,(pair)M),magenta); dot(M,UnFill); dot(D,UnFill); label("$A$",A,SW); label("$B$",B,SE); label("$M$",M,dir(midpoint(C--D)--M)); label("$C$",C,S); label("$D$",D,S); //label(slant(-.28)*"$\varphi$", (1,0), fontsize(25)+yellow); label(slant(-.28)*"$\varphi$",D--C,magenta); label("$\phantom{\varphi\,}^3$",D--C,magenta); //label("$E$",(GR,0),S); dot(A ^^B ^^C //^^M ,UnFill); //dot((GR,0), FillDraw(yellow)); /* label( "$CE/CB=\frac{\sqrt5-1}{2}=\,\mathpunct{.}618\ldots$" // "$CE/CB=\dfrac{\sqrt5-1}{2}=\,.618\ldots$" ??!? ,truepoint(2S),S); */ //label("1",C--A,fontsize(8)); point m1=midpoint(A--D); point m2=midpoint(B--D); point goldpoint_1=relpoint(M--A,gr^2); point P=goldpoint_1; point Q=intersectionpoint(M--B, Arc(m2,arclength(B--m2),1,180)); point goldpoint_2=relpoint(Q--P,gr); point R=goldpoint_2; point myS=midpoint(P--Q); point T=midpoint(A--M); draw(Arc(m1,arclength(A--m1),0,180),blue); draw(Arc(m2,arclength(B--m2),0,180),blue); draw(M--relpoint(M--A,gr^2),red+linewidth(2)); draw(A--relpoint(M--A,gr^2),blue+linewidth(2)); draw(Q--goldpoint_2,blue+linewidth(2)); draw(P--goldpoint_2,red+linewidth(2)); draw(Circle(myS, arclength(M--myS)),lightgrey); //draw(Circle(T,.5*arclength(M--A)),lightgrey); // /* usepackage("siunitx", "decimalsymbol=\mathpunct{.}\mbox{}"); // */ // /* tex(" \sisetup{ group-separator={,}, decimalsymbol=\mathpunct{.}\mbox{} } "); // */ draw(rotate(degrees(B-M))*format("$\SI{%f6}\ldots^\circ$",degrees(M-C)),Arc((pair)C,(pair)B,(pair)M),magenta); draw(B--C--M,magenta); draw("$1$",(A.x,-.3)--(C.x,-.3),LeftSide,Bars); //label("$P$",P,I*dir(A--M)); label("$P$",P,dir(C--P)); label("$R$",R,3*dir(R--relpoint(M--P,.55)),fontsize(5)); label("$S$",myS,2*dir(myS--relpoint(M--B,.3)),fontsize(5)); dot(m1^^m2,blue,UnFill); dot(goldpoint_1, FillDraw(yellow)); dot(goldpoint_2, FillDraw(yellow)); dot(A^^C^^D^^B^^M^^myS, UnFill); dot(Q,UnFill); dot(T,UnFill); label("$T$",T,1.5*I*dir(A--M),fontsize(5)); label("$Q$",Q,dir(Q--midpoint(D--m2))); //dot(relpoint(M--C,.6)); label( "$AP/PM=QR/RP=\frac{\sqrt5-1}{2}=\,0\mathpunct{.}618\ldots$" // "$CE/CB=\dfrac{\sqrt5-1}{2}=\,.618\ldots$" ??!? ,truepoint(2.5S),S); shipout(bbox(3.5mm,3.5mm,FillDraw( yellow+.8blue ,orange+.5blue +GR*mm+miterjoin) ));[/asy]$ $[asy]label("Figure 7: Mispaced and cutted Labels, because of siunitx?");[/asy]$

$[asy] import geometry; usepackage("amsmath"); size(8cm,10cm); unitsize(3.5cm); point A,B,C; real GR=(sqrt(5)+1)/2; real gr=(sqrt(5)-1)/2; A=(0,0); B=(2,0); C=(1,0); path arc_1=arc(C,1,0,180) ; draw(arc_1); /* pair M=intersectionpoint(arc_1,arc_2); path arc_4=Arc(A,(GR,0),M); draw(arc_4,magenta); draw(arc_1); //draw(arc_4); draw("$72^\circ$",arc_3,magenta+linewidth(2)); draw(C--M,magenta+linewidth(2)); */ //draw(A--M--B); /* draw(M--(M.x,0),dotted); draw(C--(GR,0),blue+linewidth(2)); draw((GR,0)--B,red+linewidth(2)); */ point D=C+gr^3; line perp=perpendicular(D, line(A,B)); //draw(perp); point Dprime=(D.x,1); //dot(Dprime); point M=intersectionpoint(D--Dprime,arc_1); draw(A--M--B^^M--D^^C--B^^M--C); /* point M=intersectionpoint(perp, arc_1); no matching function 'intersectionpoint(, path) */ //point M=intersectionpoint(arc_1,perp); ambigous draw(A--C); //draw(Arc((pair)C,(pair)B,(pair)M),magenta); dot(M,UnFill); dot(D,UnFill); label("$A$",A,SW); label("$B$",B,SE); label("$M$",M,dir(midpoint(C--D)--M)); label("$C$",C,S); label("$D$",D,S); //label(slant(-.28)*"$\varphi$", (1,0), fontsize(25)+yellow); label(slant(-.28)*"$\varphi$",D--C,magenta); label("$\phantom{\varphi\,}^3$",D--C,magenta); //label("$E$",(GR,0),S); dot(A ^^B ^^C //^^M ,UnFill); //dot((GR,0), FillDraw(yellow)); /* label( "$CE/CB=\frac{\sqrt5-1}{2}=\,\mathpunct{.}618\ldots$" // "$CE/CB=\dfrac{\sqrt5-1}{2}=\,.618\ldots$" ??!? ,truepoint(2S),S); */ //label("1",C--A,fontsize(8)); point m1=midpoint(A--D); point m2=midpoint(B--D); point goldpoint_1=relpoint(M--A,gr^2); point P=goldpoint_1; point Q=intersectionpoint(M--B, Arc(m2,arclength(B--m2),1,180)); point goldpoint_2=relpoint(Q--P,gr); point R=goldpoint_2; point myS=midpoint(P--Q); point T=midpoint(A--M); draw(Arc(m1,arclength(A--m1),0,180),blue); draw(Arc(m2,arclength(B--m2),0,180),blue); draw(M--relpoint(M--A,gr^2),red+linewidth(2)); draw(A--relpoint(M--A,gr^2),blue+linewidth(2)); draw(Q--goldpoint_2,blue+linewidth(2)); draw(P--goldpoint_2,red+linewidth(2)); draw(Circle(myS, arclength(M--myS)),mediumgrey); //draw(Circle(T,.5*arclength(M--A)),lightgrey); // /* usepackage("siunitx", "decimalsymbol=\mathpunct{.}\mbox{}"); // */ // /* tex(" \sisetup{ group-separator={,}, decimalsymbol=\mathpunct{.}\mbox{} } "); // */ draw(rotate(degrees(B-M))*format("$\SI{%f6}\ldots^\circ$",degrees(M-C)),Arc((pair)C,(pair)B,(pair)M),magenta+fontsize(10)); draw(B--C--M,magenta); draw("$1$",(A.x,-.3)--(C.x,-.3),LeftSide,Bars); //label("$P$",P,I*dir(A--M)); label("$P$",P,dir(D--P)); label("$R$",R,4*dir(R--relpoint(M--P,.75)),fontsize(5)); label("$S$",myS,.8*dir(myS--relpoint(M--B,.05)),fontsize(5)); dot(m1^^m2,blue,UnFill); dot(goldpoint_1, FillDraw(yellow)); dot(goldpoint_2, FillDraw(yellow)); dot(A^^C^^D^^B^^M^^myS, UnFill); dot(Q,UnFill); dot(T,UnFill); label("$T$",T,1.5*I*dir(A--M),fontsize(5)); label("$Q$",Q,dir(Q--midpoint(D--m2))); //dot(relpoint(M--C,.6)); label("$m_1$",m1,2S,blue+fontsize(5)); label("$m_2$",m2,2S,blue+fontsize(5)); label( "$AP/PM=QR/RP=\frac{\sqrt5-1}{2}=\,0\mathpunct{.}618\ldots$" // "$CE/CB=\dfrac{\sqrt5-1}{2}=\,.618\ldots$" ??!? ,truepoint(2.5S),S); shipout(bbox(3.5mm,3.5mm,FillDraw( yellow+.8blue ,orange+.5blue +GR*mm+miterjoin) ));[/asy]$ $[asy]label("Figure 8: Yes, because of siunitx.");[/asy]$

$DM$ and $PQ$ span up a rectangle, from which the sidelength as proportion will be of further interest. It will be possible to draw the Pappus chain into the arbelos using inversions. The midpoints of the circles therefor are said to be on an ellipse. And from these midpoints you can derive number sequences as it was made with the Golden Arbelos. The numbers we will find here - you can ask you - if these numbers already are known.

Hiroschi Okumura (2016) in 1.pdf shows regular pentagrams of different size in the Golden Arbelos. The same author also in 2016 in [url//http://geometry-math-journal.ro/pdf/Volume5-Issue1/6.pdf]6.pdf[/url] mentions some squares in the Golden Arbelos.

$[asy] // sobald fontsize: autosizing. size(8cm); real gr=(sqrt(5)-1)/2; //real my_angle_0=76.345415254024; // arccos(((sqrt(5)-1)/2)^3) real my_angle_1=aCos(gr^3); real my_angle_2=degrees(acos(gr^3)); real my_angle_1=aCos(gr^3); int my_Arc_n=400; pair A=(-1,0); pair B=(1,0); pair C=dir(my_angle_1); pair M3=midpoint(C--(C.x,0)); pair Cx=(gr^3,0); pair helpPoint_1=dir(my_angle_1); pair helpPoint_2=(gr^3,0); pair m1=midpoint(A--Cx); pair m2=midpoint(B--Cx); path Arc_0=Arc(origin,1,0,180,my_Arc_n); path Arc_1=Arc(m1,arclength(A--m1),0,180,my_Arc_n); path Arc_2=Arc(m2,arclength(B--m2),0,180,my_Arc_n); path Arc_2_help=Arc(m2,arclength(B--m2),1,180,my_Arc_n); path redCircle=Circle(M3, arclength(C--M3),my_Arc_n); pair gold_candidat_1=relpoint(A--C,gr); pair P=intersectionpoint(A--C, Arc_1); pair Q=intersectionpoint(B--C, Arc_2_help); pair Px=(P.x,0) ,Qx=(Q.x,0); fill(buildcycle(Arc_0,Arc_1,Arc_2),lightgrey); draw(P--Px^^Q--Qx,dotted); draw(redCircle,red); draw(C--Cx); draw(C--P--Cx--Q--cycle,.9green+linewidth(1)); draw(B--origin--C,magenta+linewidth(2)); draw(Arc(origin,B,C),magenta+linewidth(2)); draw(Arc(origin,C,A),linewidth(1)); draw(P--A--origin^^B--Q); draw(Arc_1,blue+linewidth(1)); draw(Arc_2,blue+linewidth(1)); import geometry; line line_3=perpendicular(C,line(origin,C)); line line_4=parallel(P,line_3); line line_5=parallel(Q,line(A,B)); draw(line_3); draw(line_4); draw(line_5); dot(A^^B^^C,UnFill); dot("$C^{\scriptscriptstyle\prime}$",Cx, 1.75*S,fontsize(5), UnFill); dot(origin,UnFill); //dot(gold_candidat_1,Fill(Yellow)); dot(P^^Px,UnFill); dot(Q^^Qx,UnFill); dot((0,1.1),invisible); dot(m1^^m2,blue,UnFill); dot(M3,red,UnFill); label("$A$", A, SW, fontsize(8)); label("$B$", B, SE, fontsize(8)); label("$C$", C, dir(origin--C), fontsize(8)); label("$M_0$", origin, S, fontsize(8)); label("$M_1$",m1,1.75*S,blue+fontsize(5)); label("$M_2$",m2,1.75*S,blue+fontsize(5)); label("$M_3$",M3,1.3*dir(M3--midpoint(C--Q)),red+fontsize(5)); label("$P$",P, 1.5dir(origin--P),fontsize(5)); label("$Q$",Q, 1.5dir(Cx--Q),fontsize(5)); // https://texfaq.org/FAQ-mathsize label("$P^{\scriptscriptstyle\prime}$",Px,1.75S,fontsize(5)); label("$Q^{\scriptscriptstyle\prime}$",Qx,1.75S,fontsize(5)); usepackage("siunitx", "locale=US, decimalsymbol=\mathpunct{.}\mbox{}, group-separator={,}"); label(rotate(degrees(B-C))*format("$\SI{%#1.12f}\ldots^\circ$",degrees(C-origin)),Arc(origin,B,C),fontsize(10)+magenta); shipout(bbox(3mm, Fill(white+.9Yellow))); [/asy]$

usepackage("siunitx", "locale=US, decimalsymbol=\mathpunct{.}\mbox{}, group-separator={,}");
label(rotate(degrees(B-C))*format("$\SI{%#1.12f}\ldots^\circ$",degrees(C-origin)),Arc(origin,B,C),fontsize(10)+magenta);

$[asy] usepackage("siunitx", "locale=US , decimalsymbol=\mathpunct{.}\mbox{} , group-separator={,}"); // sobald fontsize: autosizing. size(8cm); real gr=(sqrt(5)-1)/2; real GR=(sqrt(5)+1)/2; //real my_angle_0=76.345 415 254 024; // arccos(((sqrt(5)-1)/2)^3) real my_angle_1=aCos(gr^3); real my_angle_2=degrees(acos(gr^3)); real my_angle_1=aCos(gr^3); int my_Arc_n=5000;// AoPS accepts: 5000 pair A=(-1,0); pair B=(1,0); pair C=dir(my_angle_1); pair M3=midpoint(C--(C.x,0)); pair Cx=(gr^3,0); pair helpPoint_1=dir(my_angle_1); pair helpPoint_2=(gr^3,0); pair M1=midpoint(A--Cx); pair M2=midpoint(B--Cx); path Arc_0=Arc(origin,1,0,180,my_Arc_n); path Arc_1=Arc(M1,arclength(A--M1),0,180,my_Arc_n); path Arc_2=Arc(M2,arclength(B--M2),0,180,my_Arc_n); path Arc_2_help=Arc(M2,arclength(B--M2),1,180,my_Arc_n); pair M0=origin; path Arc_3=Arc(M0,arclength(M0--Cx),0,180); path redCircle=Circle(M3, arclength(C--M3),my_Arc_n); pair gold_candidat_1=relpoint(A--C,gr); pair P=intersectionpoint(A--C, Arc_1); pair Q=intersectionpoint(B--C, Arc_2_help); pair Px=(P.x,0) ,Qx=(Q.x,0); fill(buildcycle(Arc_0,Arc_1,Arc_2),lightgrey); draw(P--Px^^Q--Qx,dotted); draw(Arc_3,blue+linewidth(.75)); draw(redCircle,red); draw(C--Cx); draw(C--P--Cx--Q--cycle,.9green+linewidth(1)); draw(B--origin--C,magenta+linewidth(2)); draw(Arc(origin,B,C),magenta+linewidth(2)); draw(Arc(origin,C,A),linewidth(1)); draw(P--A--origin^^B--Q); draw(Arc_1,blue+linewidth(1)); draw(Arc_2,blue+linewidth(1)); import geometry; line line_3=perpendicular(C,line(origin,C)); line line_4=parallel(P,line_3); line line_5=parallel(Q,line(A,B)); ellipsenodesnumberfactor=my_Arc_n; ellipse e_1=ellipse(P,Q,C); draw(e_1,Yellow); //ellipse e_2=ellipse(P,C,Q); draw(e_2,Yellow); //ellipse e_3=ellipse(C,Q,P); draw(e_3,Yellow); draw(line_3); draw(line_4); draw(line_5); //draw(line(A,C)); //draw(line(B,M3)); dot(A^^B^^C,UnFill); dot("$C^{\scriptscriptstyle\prime}$",Cx, 1.75*S,fontsize(5), UnFill); dot(origin,UnFill); //dot(gold_candidat_1,Fill(Yellow)); dot(P^^Px,UnFill); dot(Q^^Qx,UnFill); dot((0,1.1),invisible); dot(M1^^M2,blue,UnFill); dot(M3,red,UnFill); label("$A$", A, SW, fontsize(8)); label("$B$", B, SE, fontsize(8)); label("$C$", C, dir(origin--C), fontsize(8)); label("$M_0$", origin, S, fontsize(8)); label("$M_1$",M1,1.75*S,blue+fontsize(5)); label("$M_2$",M2,1.75*S,blue+fontsize(5)); label("$M_3$",M3,1.3*dir(M3--midpoint(C--Q)),red+fontsize(5)); label("$P$",P, 1.5dir(origin--P),fontsize(5)); label("$Q$",Q, 1.5dir(Cx--Q),fontsize(5)); // https://texfaq.org/FAQ-mathsize label("$P^{\scriptscriptstyle\prime}$",Px,1.75S,fontsize(5)); label("$Q^{\scriptscriptstyle\prime}$",Qx,1.75S,fontsize(5)); label(rotate(degrees(B-C))*format("$\SI{%#4.12f}\ldots^\circ$",degrees(C-origin)),Arc(origin,B,C),fontsize(8)+magenta); string uu="%#4.12f"; label("$ \setlength{\arraycolsep}{.2em} \begin{array}[t]{ll} P&\approx(\,\SI{"+format(uu,P.x)+ "}, +\SI{"+format(uu,P.y)+"})\\ Q&\approx(+\,\SI{"+format(uu,Q.x)+ "}, +\SI{"+format(uu,Q.y)+"})\\ \end{array} $",truepoint(2.5S),S,fontsize(7)); // http://paletton.com/#uid=10G0u0khkDI7aVccCNdlmvqpOsi // 255 194 117 real c1=255/255 ,c2=194/255 ,c3=117/255 ,c4=255/255 ,c5=135/255 ,c6=94/255 ; //pen cmyk(real c, real m, real y, real k); pen my_cmyk_1=cmyk(100,76.1,45.9,43);//schwarz //shipout(bbox(3mm, Fill(white+.9Yellow))); shipout(bbox(3.5mm,3.5mm,FillDraw( rgb(c1,c2,c3)//yellow+.8blue //,orange+.5blue ,lightgrey+orange+.3blue+red //,my_cmyk_1 //,rgb(c4,c5,c6) +GR*mm+miterjoin) )); [/asy]$
$[asy] usepackage("siunitx", "locale=US , decimalsymbol=\mathpunct{.}\mbox{} , group-separator={,}"); // sobald fontsize: autosizing. size(8cm); real gr=(sqrt(5)-1)/2; real GR=(sqrt(5)+1)/2; //real my_angle_0=76.345 415 254 024; // arccos(((sqrt(5)-1)/2)^3) real my_angle_1=aCos(gr^3); real my_angle_2=degrees(acos(gr^3)); real my_angle_1=aCos(gr^3); int my_Arc_n=400;// AoPS accepts: 5000 pair A=(-1,0); pair B=(1,0); pair C=dir(my_angle_1); pair M3=midpoint(C--(C.x,0)); pair Cx=(gr^3,0); pair helpPoint_1=dir(my_angle_1); pair helpPoint_2=(gr^3,0); pair M1=midpoint(A--Cx); pair M2=midpoint(B--Cx); path Arc_0=Arc(origin,1,0,180,my_Arc_n); path Arc_1=Arc(M1,arclength(A--M1),0,180,my_Arc_n); path Arc_2=Arc(M2,arclength(B--M2),0,180,my_Arc_n); path Arc_2_help=Arc(M2,arclength(B--M2),1,180,my_Arc_n); path redCircle=Circle(M3, arclength(C--M3),my_Arc_n); pair M0=origin; path Arc_3=Arc(M0,arclength(M0--Cx),0,180,my_Arc_n); pair gold_candidat_1=relpoint(A--C,gr); pair P=intersectionpoint(A--C, Arc_1); pair Q=intersectionpoint(B--C, Arc_2_help); pair Px=(P.x,0) ,Qx=(Q.x,0); pair Pprimeprime=(P.x,P.y*GR); dot(Pprimeprime); fill(buildcycle(Arc_0,Arc_1,Arc_2),lightgrey); draw(P--Px^^Q--Qx,dotted); draw(Px--P,blue+linewidth(2)); draw(P--Pprimeprime,red+linewidth(2)); draw(redCircle,red); draw(C--Cx); draw(C--P--Cx--Q--cycle,.9green+linewidth(1)); draw(B--origin--C,magenta+linewidth(2)); draw(Arc(origin,B,C),magenta+linewidth(2)); draw(Arc(origin,C,A),linewidth(1)); draw(P--A--origin^^B--Q); draw(Arc_1,blue+linewidth(1)); draw(Arc_2,blue+linewidth(1)); draw(Arc_3,blue+linewidth(1)); import geometry; line line_3=perpendicular(C,line(origin,C)); line line_4=parallel(P,line_3); line line_5=parallel(Q,line(A,B)); draw(line(Pprimeprime,C)); ellipsenodesnumberfactor=my_Arc_n; ellipse e_1=ellipse(P,Q,C); draw(e_1,Yellow); //ellipse e_2=ellipse(P,C,Q); draw(e_2,Yellow); //ellipse e_3=ellipse(C,Q,P); draw(e_3,Yellow); draw(line_3); draw(line_4); draw(line_5); //draw(line(A,C)); //draw(line(B,M3)); dot(A^^B^^C^^Pprimeprime,UnFill); dot("$C^{\scriptscriptstyle\prime}$",Cx, 1.75*S,fontsize(5), UnFill); dot(origin,UnFill); //dot(gold_candidat_1,Fill(Yellow)); dot(P^^Px,UnFill); dot(Q^^Qx,UnFill); dot((0,1.1),invisible); dot(M1^^M2,blue,UnFill); dot(M3,red,UnFill); label("$A$", A, SW, fontsize(8)); label("$B$", B, SE, fontsize(8)); label("$C$", C, dir(origin--C), fontsize(8)); label("$M_0$", origin, S, fontsize(8)); label("$M_1$",M1,1.75*S,blue+fontsize(5)); label("$M_2$",M2,1.75*S,blue+fontsize(5)); label("$M_3$",M3,1.3*dir(M3--midpoint(C--Q)),red+fontsize(5)); label("$P$",P, 1.5dir(origin--P),fontsize(5)); label("$Q$",Q, 1.5dir(Cx--Q),fontsize(5)); // https://texfaq.org/FAQ-mathsize label("$P^{\scriptscriptstyle\prime}$",Px,1.75S,fontsize(5)); label("$Q^{\scriptscriptstyle\prime}$",Qx,1.75S,fontsize(5)); label(rotate(degrees(B-C))*format("$\SI{%#4.12f}\ldots^\circ$",degrees(C-origin)),Arc(origin,B,C),fontsize(8)+magenta); string uu="%#4.12f"; label("$ \setlength{\arraycolsep}{.2em} \begin{array}[t]{ll} P&\approx(\,\SI{"+format(uu,P.x)+ "}, +\SI{"+format(uu,P.y)+"})\\ Q&\approx(+\,\SI{"+format(uu,Q.x)+ "}, +\SI{"+format(uu,Q.y)+"})\\ \end{array} $",truepoint(2.5S),S,fontsize(7)); // http://paletton.com/#uid=10G0u0khkDI7aVccCNdlmvqpOsi // 255 194 117 real c1=255/255 ,c2=194/255 ,c3=117/255 ,c4=255/255 ,c5=135/255 ,c6=94/255 ; //pen cmyk(real c, real m, real y, real k); pen my_cmyk_1=cmyk(100,76.1,45.9,43);//schwarz //shipout(bbox(3mm, Fill(white+.9Yellow))); shipout(bbox(3.5mm,3.5mm,FillDraw( rgb(c1,c2,c3)//yellow+.8blue //,orange+.5blue ,lightgrey+orange+.3blue+red //,my_cmyk_1 //,rgb(c4,c5,c6) +GR*mm+miterjoin) )); [/asy]$
$[asy] usepackage("siunitx", "locale=US , decimalsymbol=\mathpunct{.}\mbox{} , group-separator={,}"); // sobald fontsize: autosizing. size(8cm); real gr=(sqrt(5)-1)/2; real GR=(sqrt(5)+1)/2; //real my_angle_0=76.345 415 254 024; // arccos(((sqrt(5)-1)/2)^3) real my_angle_1=aCos(gr^3); real my_angle_2=degrees(acos(gr^3)); real my_angle_1=aCos(gr^3); int my_Arc_n=400;// AoPS accepts: 5000 pair A=(-1,0); pair B=(1,0); pair C=dir(my_angle_1); pair Cx=(gr^3,0); //pair C=( Cx.x,sqrt(1-(gr^3)^2)); pair M3=midpoint(C--(C.x,0)); pair helpPoint_1=dir(my_angle_1); pair helpPoint_2=(gr^3,0); pair M1=midpoint(A--Cx); pair M2=midpoint(B--Cx); path Arc_0=Arc(origin,1,0,180,my_Arc_n); path Arc_1=Arc(M1,arclength(A--M1),0,180,my_Arc_n); path Arc_2=Arc(M2,arclength(B--M2),0,180,my_Arc_n); path Arc_2_help=Arc(M2,arclength(B--M2),1,180,my_Arc_n); path redCircle=Circle(M3, arclength(C--M3),my_Arc_n); pair M0=origin; path Arc_3=Arc(M0,arclength(M0--Cx),0,180,my_Arc_n); pair gold_candidat_1=relpoint(A--C,gr); pair P=intersectionpoint(A--C, Arc_1); pair Q=intersectionpoint(B--C, Arc_2_help); pair Px=(P.x,0) ,Qx=(Q.x,0); pair Pprimeprime=(P.x,P.y*GR); dot(Pprimeprime); fill(buildcycle(Arc_0,Arc_1,Arc_2),lightgrey); draw(//P--Px^^ Q--Qx,dotted); draw(A--P^^P--Px,blue+linewidth(2)); draw(P--C^^P--(P.x,C.y),red+linewidth(2)); draw(redCircle,red); draw(C--Cx); draw(P--Cx--Q--C,.9green+linewidth(1)); draw(B--origin--C,magenta+linewidth(2)); draw(Arc(origin,B,C),magenta+linewidth(2)); draw(Arc(origin,C,A),linewidth(1)); draw(A--origin^^B--Q); draw(Arc_1,blue+linewidth(1)); draw(Arc_2,blue+linewidth(1)); draw(Arc_3,blue+linewidth(1)); import geometry; line line_3=perpendicular(C,line(origin,C)); line line_4=parallel(P,line_3); line line_5=parallel(Q,line(A,B)); draw(line(Pprimeprime,C)); ellipsenodesnumberfactor=my_Arc_n; ellipse e_1=ellipse(P,Q,C); draw(e_1,Yellow); //ellipse e_2=ellipse(P,C,Q); draw(e_2,Yellow); //ellipse e_3=ellipse(C,Q,P); draw(e_3,Yellow); draw(line_3); draw(line_4); draw(line_5); //draw(line(A,C)); //draw(line(B,M3)); dot(A^^B^^C^^Pprimeprime,UnFill); dot("$C^{\scriptscriptstyle\prime}$",Cx, 1.75*S,fontsize(5), UnFill); dot(origin,UnFill); //dot(gold_candidat_1,Fill(Yellow)); dot(P^^Px,UnFill); dot(Q^^Qx,UnFill); dot((0,1.1),invisible); dot(M1^^M2,blue,UnFill); dot(M3,red,UnFill); label("$A$", A, SW, fontsize(8)); label("$B$", B, SE, fontsize(8)); label("$C$", C, dir(origin--C), fontsize(8)); label("$M_0$", origin, S, fontsize(8)); label("$M_1$",M1,1.75*S,blue+fontsize(5)); label("$M_2$",M2,1.75*S,blue+fontsize(5)); label("$M_3$",M3,1.3*dir(M3--midpoint(C--Q)),red+fontsize(5)); label("$P$",P, 1.5dir(origin--P),fontsize(5)); label("$Q$",Q, 1.5dir(Cx--Q),fontsize(5)); // https://texfaq.org/FAQ-mathsize label("$P^{\scriptscriptstyle\prime}$",Px,1.75S,fontsize(5)); label("$Q^{\scriptscriptstyle\prime}$",Qx,1.75S,fontsize(5)); label("$P^{\scriptscriptstyle\prime\mkern-1.5mu\prime}$" ,(P.x,C.y),1.25*dir(85),fontsize(5)); label(rotate(degrees(B-C))*format("$\SI{%#4.12f}\ldots^\circ$",degrees(C-origin)),Arc(origin,B,C),fontsize(8)+magenta); //dot(relpoint(Px--(P.x,C.y),gr)); pair grP=gold_candidat_1; pair grP2=relpoint(Px--(P.x,C.y),gr); string uu="%#4.12f"; label("$ \setlength{\arraycolsep}{.2em} \begin{array}[t]{rl} P&\approx(\,\SI{"+format(uu,P.x)+ "}, +\SI{"+format(uu,P.y)+"})\\ Q&\approx(+\,\SI{"+format(uu,Q.x)+ "}, +\SI{"+format(uu,Q.y)+"})\\ C&\approx(+\,\SI{"+format(uu,C.x)+ "}, +\SI{"+format(uu,C.y)+"})\\ grP_1&\approx(\,\SI{"+format(uu,grP.x)+ "}, +\SI{"+format(uu,grP.y)+"})\\ grP_2&\approx(\,\SI{"+format(uu,grP2.x)+ "}, +\SI{"+format(uu,grP2.y)+"})\\ \end{array} $",truepoint(2.5S),S,fontsize(7)); // grP2 // http://paletton.com/#uid=10G0u0khkDI7aVccCNdlmvqpOsi // 255 194 117 real c1=255/255 ,c2=194/255 ,c3=117/255 ,c4=255/255 ,c5=135/255 ,c6=94/255 ; //pen cmyk(real c, real m, real y, real k); pen my_cmyk_1=cmyk(100,76.1,45.9,43);//schwarz //shipout(bbox(3mm, Fill(white+.9Yellow))); shipout(bbox(3.5mm,3.5mm,FillDraw( rgb(c1,c2,c3)//yellow+.8blue //,orange+.5blue ,lightgrey+orange+.3blue+red //,my_cmyk_1 //,rgb(c4,c5,c6) +GR*mm+miterjoin) )); [/asy]$
$[asy] usepackage("siunitx", "locale=US , decimalsymbol=\mathpunct{.}\mbox{} , group-separator={,}"); // sobald fontsize: autosizing. size(8cm); real gr=(sqrt(5)-1)/2; real GR=(sqrt(5)+1)/2; //real my_angle_0=76.345 415 254 024; // arccos(((sqrt(5)-1)/2)^3) real my_angle_1=aCos(gr^3); real my_angle_2=degrees(acos(gr^3)); real my_angle_1=aCos(gr^3); int my_Arc_n=400;// AoPS accepts: 5000 pair A=(-1,0); pair B=(1,0); pair C=dir(my_angle_1); pair Cx=(gr^3,0); //pair C=( Cx.x,sqrt(1-(gr^3)^2)); pair M3=midpoint(C--(C.x,0)); pair helpPoint_1=dir(my_angle_1); pair helpPoint_2=(gr^3,0); pair M1=midpoint(A--Cx); pair M2=midpoint(B--Cx); path Arc_0=Arc(origin,1,0,180,my_Arc_n); path Arc_1=Arc(M1,arclength(A--M1),0,180,my_Arc_n); path Arc_2=Arc(M2,arclength(B--M2),0,180,my_Arc_n); path Arc_2_help=Arc(M2,arclength(B--M2),1,180,my_Arc_n); path redCircle=Circle(M3, arclength(C--M3),my_Arc_n); pair M0=origin; path Arc_3=Arc(M0,arclength(M0--Cx),0,180,my_Arc_n); pair P=intersectionpoint(A--C, Arc_1); pair Q=intersectionpoint(B--C, Arc_2_help); pair Px=(P.x,0),Qx=(Q.x,0); pair gold_candidat_1=relpoint(A--C,gr); pair grP1=gold_candidat_1; pair grP2=relpoint(Px--(P.x,C.y),gr); pair relpoint_phi_cubic_dirC=relpoint(M0--C,gr^3); dot(relpoint_phi_cubic_dirC); pair Pprimeprime=(P.x,P.y*GR); dot(Pprimeprime); fill(buildcycle(Arc_0,Arc_1,Arc_2),lightgrey); draw(//P--Px^^ Q--Qx,dotted); draw(A--P^^P--Px,blue+linewidth(2)); draw(P--C^^P--(P.x,C.y),red+linewidth(2)); draw(redCircle,red); draw(C--Cx); draw(P--Cx--Q--C,.9green+linewidth(1)); draw(B--origin--C,magenta+linewidth(2)); draw(Arc(origin,B,C),magenta+linewidth(2)); draw(Arc(origin,C,A),linewidth(1)); draw(A--origin^^B--Q); draw(Arc_1,blue+linewidth(1)); draw(Arc_2,blue+linewidth(1)); draw(Arc_3,blue+linewidth(1)); import geometry; line line_3=perpendicular(C,line(origin,C)); line line_4=parallel(P,line_3); line line_5=parallel(Q,line(A,B)); line line_6=parallel(relpoint_phi_cubic_dirC,line(P,Q)); draw(line(Pprimeprime,C)); ellipsenodesnumberfactor=my_Arc_n; ellipse e_1=ellipse(P,Q,C); draw(e_1,Yellow); //ellipse e_2=ellipse(P,C,Q); draw(e_2,Yellow); //ellipse e_3=ellipse(C,Q,P); draw(e_3,Yellow); draw(line_3); draw(line_4); draw(line_5); draw(line_6); dot(A^^B^^C^^Pprimeprime,UnFill); dot("$C^{\scriptscriptstyle\prime}$",Cx, 1.75*S,fontsize(5), UnFill); dot(origin,UnFill); //dot(gold_candidat_1,Fill(Yellow)); dot(P^^Px,UnFill); dot(Q^^Qx,UnFill); dot((0,1.1),invisible); dot(M1^^M2,blue,UnFill); dot(M3,red,UnFill); dot(relpoint_phi_cubic_dirC,UnFill); label("$A$", A, SSW, fontsize(8)); label("$B$", B, SSE, fontsize(8)); label("$C$", C, dir(origin--C), fontsize(8)); label("$M_0$", origin, S, fontsize(8)); label("$M_1$",M1,1.75*S,blue+fontsize(5)); label("$M_2$",M2,1.75*S,blue+fontsize(5)); label("$M_3$",M3,1.3*dir(M3--midpoint(C--Q)),red+fontsize(5)); label("$P$",P, 1.5dir(origin--P),fontsize(5)); label("$Q$",Q, 1.5dir(Cx--Q),fontsize(5)); label("$P^{\scriptscriptstyle\prime}$",Px,1.75S,fontsize(5)); label("$Q^{\scriptscriptstyle\prime}$",Qx,1.75S,fontsize(5)); label("$P^{\scriptscriptstyle\prime\mkern-1.5mu\prime}$" ,(P.x,C.y),1.25*dir(85),fontsize(5)); label(slant(-.28)*"$\varphi$" ,relpoint(M0--Px,.7),.7N,fontsize(7)); // /* label("$\phantom{\varphi\,}^{ \raisebox{2pt}{\kern1.5pt 3} } $" ,relpoint(M0--Px,.7),.7N,fontsize(5)); // */ label("$C^{\scriptscriptstyle\prime\mkern-1.5mu\prime}$" ,relpoint_phi_cubic_dirC //,2*dir(midpoint(Px--M0)) ,dir(SW),fontsize(5)); label(rotate(degrees(B-C))*format("$\SI{%#4.12f}\ldots^\circ$",degrees(C-origin)),Arc(origin,B,C),fontsize(8)+magenta); //dot(relpoint(Px--(P.x,C.y),gr)); string uu="%#4.12f"; label("$ \setlength{\arraycolsep}{.2em} \begin{array}[t]{rl} C&\approx(+\,\SI{"+format(uu,C.x)+ "}, +\SI{"+format(uu,C.y)+"})\\ P&\approx(-\,\SI{"+format(uu,-P.x)+ "}, +\SI{"+format(uu,P.y)+"})\\ Q&\approx(+\,\SI{"+format(uu,Q.x)+ "}, +\SI{"+format(uu,Q.y)+"})\\ %grP_1&\approx(-\,\SI{"+format(uu,-grP1.x)+ "}, %+\SI{"+format(uu,grP1.y)+"})\\ %grP_2&\approx(-\,\SI{"+format(uu,-grP2.x)+ "}, %+\SI{"+format(uu,grP2.y)+"})\\ \end{array} $",truepoint(2.5S),S,fontsize(7)); // grP2 // http://paletton.com/#uid=10G0u0khkDI7aVccCNdlmvqpOsi // 255 194 117 real c1=255/255 ,c2=194/255 ,c3=117/255 ,c4=255/255 ,c5=135/255 ,c6=94/255 ; //pen cmyk(real c, real m, real y, real k); pen my_cmyk_1=cmyk(100,76.1,45.9,43);//schwarz //shipout(bbox(3mm, Fill(white+.9Yellow))); shipout(bbox(3.5mm,3.5mm,FillDraw( rgb(c1,c2,c3)//yellow+.8blue //,orange+.5blue ,lightgrey+orange+.3blue+red //,my_cmyk_1 //,rgb(c4,c5,c6) +GR*mm+miterjoin) )); [/asy]$
The figure $APQB$ builds an asymmetric trapezoid.