La geometria degli origami e la loro applicazione

[materia]

Salve a tutti, dovrei preparare un progetto riguardante gli origami e le loro applicazioni in campo matematico, ma non riesco a comprendere il materiale che trovo online... mi sembra spiegato da cani.
Ad esempio una delle loro applicazioni è come risolvere le equazioni di terzo grado, ecco il testo:

Now, let's solve the cubic equation x3+ax2+bx+c=0 with origami. Let two points P1 and P2 have the coordinates (a,1) and (c,b), respectively. Also let two lines L1 and L2 have the equations y+1=0 and x+c=0, respectively. Fold a line placing P1 onto L1 and placing P2 onto L2, and the slope of the crease is the solution of x3+ax2+bx+c=0.

I will explain why. Let p1 be a parabola having the focus P1 and the directrix L1. Since the crease is not parallel to the y-axis, we can let the crease have the equation y=tx+u. Let the crease be tangent to p1 at (x1,y1), and (x1-a)2=4y1. Because the crease has the equation (x1-a)(x-x1)=2(y-y1), we get t=(x1-a)/2 and u=y1-x1(x1-a)/2. From these equations, we get u=-t2-at.

When c is not 0, let p2 be a parabola having the focus P2 and the directrix L2. Let the crease be tangent to p2 at (x2,y2), and (y2-b)2=4cx2. Because the crease has the equation (y2-b)(y-y2)=2c(x-x2), we get t=2c/(y2-b) and u=y2-2cx2/(y2-b). From these equations, we get u=b+c/t. (t is not 0 because the crease is not parallel to the x-axis.) Therefore, t3+at2+bt+c=0.

When c is 0, P2 is on L2. So, either the crease is perpendicular to L2 or the crease passes through P2. In the former case, t=0. In the later case, u=b, and t2+at+b=0. Therefore t3+at2+bt+c=0.

Non capisco da dove tira fuori (x1-a)2=4y1 e che (x1-a)(x-x1)=2(y-y1) e gli analoghi identici passagi che ripete con l'altra parabola, ossia (y2-b)2=4cx2 e (y2-b)(y-y2)=2c(x-x2)

[materia]

Son riuscito a rispondermi a mezza domanda, ossia la prima uguaglianza l'ho verificata: (x1-a)^2=4y1 .
ma questa non capisco da dove sbuca (x1-a)(x-x1)=2(y-y1)

[materia]

Dopo varie ore di sclero son riuscito a risolvere il quesito, grazie comunque per chi ci avesse provato