The Unitcircle and the Golden Circle

by Klaus-Anton, May 11, 2020, 9:43 AM

We draw the unitcircle around $A$ with radius $r=1$ . At the East-Pol of the unitcircle we set point $B$ . Around B we draw a circle with the radius of the Golden Ratio, $r=\frac{\sqrt5+1}{2}=1.618\dots$ . Where do the both circles intersect?
$[asy] size(8cm); unitsize(.8cm); pair A,B=(1,0); real Golden_Ratio=(sqrt(5)+1)/2; real golden_ratio=(sqrt(5)-1)/2; //draw(circle(A,1)); draw(unitcircle); path Golden_Circle=circle(B,Golden_Ratio); draw(Golden_Circle); pair[] C=intersectionpoints(unitcircle,Golden_Circle); pair C_1=C[0], C_2=C[1]; pair D=(-golden_ratio,0); pair my_E=midpoint(A--D); draw(D--C_1--A--cycle); draw("$72^\circ$",arc(D,golden_ratio,0,72),LeftSide,blue); draw("$1.618\dots$",B--C_1,gray); draw(//"$1$", B--A,gray); draw(//"$36^\circ$", arc(B,.5,180-36,180) //,.8*LeftSide ,red //+fontsize(8) ); dot(A^^B^^C_1^^C_2^^D^^my_E); label("$A$",A,SE); label("$B$",B,E); label("$C_1$",C_1,dir(A--C_1)); label("$C_2$",C_2,dir(A--C_2)); label("$D$",D,W); label("$E$",my_E,S); label("$1$", relpoint(A--B,golden_ratio^2),N,gray); label("$36^\circ$",relpoint(arc(B,.5,180-36,180),.6), dir(relpoint(arc(B,.5,180-36,180),.6)--B),red+fontsize(8)); [/asy]$
$[asy] import x11colors; label(minipage("\textbf{\textit{Figure 1:}} \hspace{-2pt}\textit{Unitcircle and Golden Circle}%\\ %\phantom{\textbf{\textit{Figure 3:}}} \color{blue}{Regular In-Pentagon} \color{black}{of Circumcircle of the} \color{red}Cardioid %\color{black}\\ %\phantom{\textbf{\textit{Figure 3:}}} Tangent to two Points ",width=7.3cm)); shipout(bbox(.5*3mm, FillDraw(white,cyan))); [/asy]$
The unitcircle can be described with $x^2+y^2=1$ .

We follow:
$(x+1)^2+y^2=\left(\frac{\sqrt5+1}{2}\right)^2$
$(x+1)^2+(1-x^2)=\left(\frac{\sqrt5+1}{2}\right)^2$
$x^2+2x+1-1-x^2=\left(\frac{\sqrt5+1}{2}\right)^2$
$x+1=\frac{\left(\frac{\sqrt5+1}{2}\right)^2-1}{2}$

The point $E$ has $x=-0.5*\frac{\sqrt5-1}{2}=-\frac{\sqrt5-1}{4}$ .

$[asy] import x11colors; label(minipage("\textbf{\textit{Figure 2:}} \hspace{-2pt}\textit{Halfed Cardioid (with Arbelos)}\\ \phantom{\textbf{\textit{Figure 2:}}} Animation %\color{black}\\ %\phantom{\textbf{\textit{Figure 3:}}} Tangent to two Points ",width=7.5cm)); shipout(bbox(.5*3mm, FillDraw(white,cyan))); [/asy]$
I had made this animation in 2013 as a pdf and now i have converted it into the gif-format as decribed in AoPS Community > LaTeX and Asymptote > Asymptote animations. (Move with the mouse-pointer over this animation, and click the right mouse-bottom. You can now choose the option "display graphic", so you will get it enlarged and better to read.)

If you look in the animation on the movement of the point $P$ - which walks on the semiincircle of the halfed cardioid from the right to the left - and take attention how the displayed indicated length $AP$ changes, you will see it passes the Golden Ratio.

And it does it do so surprisingly exactly, that all here shown decimal places are right. Enormously impressionating. In the generation of the animation i only used whole successive rational numbers for the positionings of the relpoints of the pathlength of the halfed cardioid. And this leads me to the Golden Ratio, which is known as the most irrational number.

The length $AP$ also passes $1.414\dots=\sqrt2$ .