Jesús de la Peña Hernández

t

d

xF c

2

= −

2

a

xB =

2

d

yF=

2

at

yB=b−

On the other hand,

B F

F B

y y

x x

t

−

−

= , i.e.:

bt at dt

ct d at

at d

b

a

t

d

c

t

− −

− −

=

− −

− −

= 2

2

2

2 2

2 2

2 bt^2 −at^3 −dt^2 = 2 ct−d−at

at^3 +()()d− 2 bt^2 + 2 c−at−d= 0 (2)

Comparing equations (1) and (2), we have:

(^) a= 1 ; d− 2 b=p ; 2 c−a=q ; d=−r
(^) a= 1 ;
2
r p
b
− −
= ;
2

• 
  

  1

  
  

  q
  c ; d=−r
  Now we have available the four values a,b,c,d, i.e., we have the coordinates of C and D; that
  enables us to fold simultaneously these points over the axes OY and OX: the folding line that
  bears to t = tgα is the solution of (1). You may ease folding BF, now marking axes and points
  by transparency, now mountain-folding CD previously.
  7.12 COMPLETE EQUATION OF 4th DEGREE: ITS RESOLUTION
  It ́s not our intention to explore in detail this subject now. We ́ll only say that this equation can
  be transformed in a complete 3rd degree one by a variable change and the application of Car-
  dano ́s transformation. Then Point 7.11 may be employed. The reader can refer to any treatise
  on equations general theory.
  Y
  B
  C(a,b)
  D(c,d)
  X
  F
  O G
  A
  2