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<h4>

   Problem 4 from Quiz 2 of Wildberger's YouTube lectures on Rational Trigonometry
  </h4>

<p><b>Problem:</b>
  Suppose Q is a (convex) quadliateral with sides 5, 7, 8 and 9.  Suppose also that the spread of the diagonals is 3/4.   Find the diagonals of the quadrilateral.  Generalize.
  </p>
<a href="https://www.youtube.com/watch?v=IKE8a9W2Czk&list=PL3C58498718451C47&index=65&t=514s" target="_blank">WildTrig Video 64</a>
<p>
  <b>Solution:</b>  As it turns out there is no convex quadrilateral with the given sides and and a diagonal spread of 3/4.  The minimum diagonal spread of a quadrilateral with sides 5,7,8,9 is approximately .7709, representing an acute angle of approximately 61.04 degrees.  By changing the 5 to a 6, we can surmise from the Sage interact below that there is a unique quadrilateral with diagonal spread 3/4, although we don't know the <em>exact</em> quadrilateral.
  </p>
<p>
  <b>A non convex quadrilateral with sides 5,7,8,9 and diagonal spread of 3/4.</b>
<br> If the quadrilateral is allowed to be non convex, then there are two solutions. The diagonals $AC$ and $BD$ are shown in red.
  <table>
    <tr><td><img src="nonconvexsolution1.png" alt="One solution."></td>
      <td> <img src="nonconvexsolution2.png" alt="One solution."></td>

    </tr>
  </table>
</p>
<p>
  <b>Question:</b><em> Find the quadrilaterals with rational diagonal spread whose sides have integer length.</em>
<br>There are some easy examples -- Any kite with sides of integer length. Also, <em>any</em> quadrilateral with integer sides can be flexed to an infinite number of positions so that the diagonal spread is rational.  <b>Note:</b>  <a href="Question.html" target='blank'>Partial solution.</a>   
  </p>

<p>
<b>A generalization of problem 4:</b>
  <em>Given the consecutive sides $a,b,c,d$ of a quadrilateral and the spread of the diagonals, determine the quadrilateral.</em></p>


<b>Discussion:</b>
As is remarked in the lecture, it is easy to prove that all you need is the 4 sides and 1 diagonal to completely determine the quadrilateral.   You can also get by with the 4 sides and the angle between two adjacent sides.  But, in general, four sides and the spread of the diagonals do not suffice, as you can see by experimenting with the Sage interact below.


<p>
<b>Setup:</b> Label the vertices of the quadrilateral $A,B,C,D$ in counterclock order with $AB=a$ at the <em>bottom</em>, $BC=b$ on the <em>right</em>, $CD=c$ at the <em>top</em> and $DA=d$ on the <em>left</em>. By relabeling, we can assume that $a\ge d \ge b$ and $a\ge c$. By scaling, normalize the quadrilateral so that $a=1$. Set up a coordinate system so that $A=(0,0)$ and $B=(1,0)$ and $C=(x_1,y_1)$ and $D=(x_2,y_2)$ lie above or on the $x$-axis.
</p>
<p>
We can imagine the quadrilateral flexing as $C$ moves along a circle of radius $b$ about $B$, starting at a triangle with sides of length $a=1$, $b+c$ (if $b+c \lt a+d$ ), and $d$ or a segment of length $a+d$ (if $b+c=a+d$) and ending at a triangle with sides $a$, $b$, and $c+d$ (if $b \lt d$) or sides $a+b$, $c$, and $d$ (if $a+b\lt c+d$) or a segment of length $a+b$ (if $a+b=c+d$). Note that $a\lt b+c+d</p>

This motion is controlled by the position of $x_1$, the first coordinate of $C$, for we can express $y_1, x_2, y_2$ in terms of $x_1$: $y_1=\sqrt{b^2-(1-x_1)^2}$, $x_2=(w\,z - \sqrt {disc})/(1+z^2)$ where $z=x_1/y_1$, $w=(d^2-c^2+b^2-1+2\,x_1)/(2\,y_1)$, and $disc= d^2\,(1+z^2)-w^2$, and $y_2=\sqrt{d^2-x_2^2}$.  We can compute the leftmost and rightmost positions of $x_1$, call them $lbd$ and $ubd$ respectively.  In the interact, the variable $t\in [0,1]$ sets $x_1=(1-t)\,lbd + t\,ubd$ and computes the quadrilateral.



<br><br> <b>Instructions:</b> Click on the Start button to get started.

<br><br> 1)You can change any or all of the values for the left or right side and top.
<br><br> 2) At the Start, an animation of the possible positions of the quadrilateral with the given sides and top is shown.
 <br><br>3) To see the quadrilaterals with a given spread, put that value in the spread box and put a -1 in the $t$ box. Then press Update.(At the bottom)
 <br><br>4) To see the quadrilateral at position $t$ (where $t=0$ and $t=1$ are the extreme positions), put that value in the $t$ box, and put a -1 in the spread box.  Then press Update.




<div id="mycell"><script type="text/x-sage">

class quad(object):
    def __init__(self,b=1,c=1,d=1,s=0,graphs=true):
        var('x1')
        a=1
        A=vector([0,0]);B=vector([a,0])
        self.A=A;self.B=B;
        self.a=a;self.b=b;self.c=c;self.d=d
        y1=lambda x1:sqrt(b^2-(x1-a)^2)

        r=lambda  x1 :1/2*(d^2- c^2+x1^2+y1(x1)^2 )
        #r=lambda x1: 1/2*(d^2-c^2-1+2*x1)
        w=lambda x1:r(x1)/y1(x1)
        z=lambda x1:x1/y1(x1)
        disc=lambda x1:d^2*(1+z(x1)^2)-w(x1)^2
        xp=lambda  x1 :(w(x1)*z(x1)+sqrt(disc(x1)))/(1+z(x1)^2)
        yp=lambda  x1 :(r(x1)-x1*xp(x1))/y1(x1)
        xm=lambda  x1 :(w(x1)*z(x1)-sqrt(disc(x1)))/(1+z(x1)^2)
        ym=lambda  x1 :(r(x1)-x1*xm(x1))/y1(x1)
        self.y1=y1;self.r=r;self.w=w;self.z=z;self.disc=disc
        self.xp=xp;self.yp=yp;
        self.xm=xm;self.ym=ym
        C=lambda x1 :vector([x1,y1(x1)]);
        Dp=lambda x1 :vector([xp(x1),yp(x1)])
        Dm=lambda x1 :vector([xm(x1),ym(x1)])
        Ep= lambda x1: yp(x1)/(y1(x1)*(1-xp(x1))+x1*yp(x1))*C(x1)
        Em= lambda x1: ym(x1)/(y1(x1)*(1-xm(x1))+x1*ym(x1))*C(x1)
        self.C=C;self;self.Dp=Dp;self.Dm=Dm;self.Ep=Ep;self.Em=Em
        x2=(1+d^2-(c+b)^2)/2
        lbd=max(1+b/(c+b)*(x2-1),1-c-b)
        ubd=min((1+b),(1+(d+c)^2-b^2)/2)
        self.lbd=lbd;self.ubd=ubd
        def spread(V1,V2):
            ass=V1*V1
            bs=V2*V2
            cs=(V1-V2)*(V1-V2)
            return 1-((ass+bs-cs)^2/(4*ass*bs))
        def spread(V1,V2):
            return det(matrix([V1,V2]))^2/((V1*V1)*(V2*V2))
        sE=lambda t: spread(C(xk(t))-A,Dm(xk(t))-B)
        sA=lambda t: spread(Dm(xk(t)),B)
        sB=lambda t: spread(-B,C(xk(t))-B)
        self.spread=spread;self.sE=sE;self.sA=sA;self.sB=sB


        if graphs:
            self.G=line([A,B])+arc(B,b,sector=(0,pi),linestyle='dotted')+arc(A,d,sector=(0,pi),linestyle='dotted')
            self.Gp=lambda x1,color='blue': polygon([A,B,C(x1),Dp(x1)],fill=false,color=color)
            self.Gm=lambda x1,color='red': polygon([A,B,C(x1),Dm(x1)],fill=false,color=color)
            self.diagp=lambda x1,color='blue':line([A,C(x1)],color=color)+line([B,Dp(x1)], color=color)
            self.diagm=lambda x1,color='red':line([A,C(x1)],color=color)+line([B,Dm(x1)], color=color)
            xk=lambda t:(t*(ubd-10^5*sys.float_info.epsilon)+(1-t)*(lbd+10^5*sys.float_info.epsilon))
            self.xk=xk
            self.fm=lambda x1,color='blue':self.G+point([C(x1),Dm(x1)])+polygon([A,Dm(x1),C(x1),B],color=color,alpha=.1)
            self.fp=lambda x1,color='red':self.G+point([C(x1),Dp(x1)])+polygon([A,Dp(x1),C(x1),B],color=color,alpha=.1)
            self.hm=lambda t,colors=['blue','blue','red']:self.fm(xk(t),color=colors[0])+self.Gm(xk(t),color=colors[1])+self.diagm(xk(t),color=colors[2])+polygon([A,B,Em(xk(t))],alpha=.2,color='red',aspect_ratio=1)
            self.hp=lambda t:self.fp(xk(t))+self.Gp(xk(t))+self.diagp(xk(t),'red')+point(Ep(xk(t)))
            if s>0:
                ss=cos(s/180*pi.n())^2
                tsp=sqrt((1+sqrt(1-ss))/(2*ss)-1/4)
                tsm=sqrt((1-sqrt(1-ss))/(2*ss)-1/4)
                rsp=(tsp^2-1/4)/(2*tsp)
                rsm=(tsm^2-1/4)/(2*tsm)

                xs=cs+sqrt(cs)*sqrt(cs+1-b^2)
                Cs=C(xs);Dsp=Dp(xs);Dsm=Dm(xs)
                self.ss=cs
                self.Psp=polygon([A,B,Cs,Dsp],alpha=.2)
                self.Psm=polygon([A,B,Cs,Dsm],alpha=.2,color='red')

@interact(layout={'top':[['d','c','b','t2','sprd']]})
def quadragon(d=input_box((8/9),width=6,label='left side'),c=input_box((7/9),width=6,label='top'),b=input_box((5/9),width=6,label='right side'),t2=input_box(-1,width=6,label='t'),sprd=input_box(-1,width=6,label='spread'),auto_update=false):
    P=quad(b= b,c= c,d= d,s=0)
    G=P.G;Gm=P.Gm;Gp=P.Gp;Ep=P.Ep;Em=P.Em;A=P.A;B=P.B;C=P.C;Dm=P.Dm;Dp=P.Dp;
    ubd=P.ubd;lbd=P.lbd;diagp=P.diagp;diagm=P.diagm;xk=P.xk;sB=P.sB;sA=P.sA;sE=P.sE
    hm=P.hm
    if b>d or max([b,c,d])>1 or b+c+d<=1:
        print("Left side >= right side, the sides and top <= 1 and sum of sides and top > 1")
        return
    try:
        mx=find_root(derivative(sE(t),t),0,1)
        mx=sE(mx)
    except:
        mx=max([sE(i/100) for i in range(101)])
    mn=min([sE(0),sE(1)])
    if  not t2==-1 and not sprd==-1:
        print('Choose just one of:')
        print('1) Enter a value for t in [0,1] and the value spread = 1')
        print('2) Enter a value for spread in (0,1] and the value t = -1')
        return
    if  t2==-1 and sprd==-1:
        print('Minimum spread: '+str(mn.n(digits=6))+' Maximum spread: '+str(mx.n(digits=6)))
        print('Generating positions ...')
        show(animate([hm(1-i/20) for i in range(0,21)],figsize=4,title='Positions'))
        show(plot(sE,(0,1),title='Spread', figsize=4))
        return
    if (sprd<mn and not sprd== -1) or sprd > mx:
        print('No quadrilateral with these sides and spread.')
        show(plot(sE,(0,1))+line([(0,sprd),(1,sprd)],color='red',linestyle='dotted', figsize=3))
        return
    if not t2==-1 and t2<0 or t2>1:
        print("the value of t must be -1 or between 0 and 1.")
        return
    if not t2==-1 and t2>=0 and t2<=1:
        show(hm(t2),figsize=4)
        show(plot(sE,(0,1))+point([t2,sE(t2)],figsize=3))
        return
    var('t1')
    rts=[]
    try:
        r1=find_root(P.sE(t1)-sprd,0,1)
        rts+=[r1]
    except:
        r1=1
    try:
        r2=find_root(P.sE(t1)-sprd,0,r1-.001)
        rts+=[r2]
    except:
        2==2
    if len(rts)==1:
        print('Unique quadrilateral with this spread and sides.')
        show(hm(rts[0]),figsize=4)
        show(plot(sE,(0,1))+point([rts[0],sE(rts[0])]),figsize=4)
        return
    if len(rts)==2:
        strt='Two '
        if b==d:
            strt+=' congruent '
        print(strt+'quadrilaterals with this spread and sides.')
        show(hm(rts[0])+hm(rts[1]),figsize=4)
        show(plot(sE,(0,1))+point([(rts[0],sE(rts[0])),(rts[1],sE(rts[1]))]),figsize=3)
        return
  </script>
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Collaborative Calculation and Data Science

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A Sagemath interact to investigate quadrilaterals.
