Jesús de la Peña Hernández
9.10 TRISECTION OF ANY ANGLE (H. HUZITA)
Let AIB = 3α the angle to trisect.
In Point 7.14.5 this equation was solved:
t^3 +t^2 − 2 t− 1 = 0 (3)
Similarities and dissimilarities between (2) and (3):
- Both are complete equations of third degree.
- With negative discriminants (ignore in (2) the values of
2
3
π
α=n , n being a natural
number; these exceptions will be studied below).
- Every coefficient of (3) can be expressed in the same unit.
- (2), on the contrary, requires two different types of unit to express its coefficients:
the unit as such, like in (3), for the monomial terms of third and first degree, and the unit a
for the terms of second degree and independent. - Fortunately, coefficients using as unit 1 and a, alternate in (2): so, all the horizontal
vectors will be measured with a type of unit and the vertical ones, with the other.
Before proceeding, let ́s solve (2) for 3α equal to 90º and 180º respectively.
Being a = tg 90º = ∞, let divide (2) by a:
− 3 t^2 + 1 = 0 ;
3
1
t= ; 30 º
3
1
α=Arctg =
As a = tg 180º = 0, (2) becomes:
t^3 − 3 t= 0 ; t^2 − 3 = 0 ; t= 3 ; α=Arctg 3 = 60 º
It ́s evident that these particular solutions are direct: 30
3
90
= ; 60
3
180
=
In Point 8.2.2.2 we already studied the obtention of angles of 30º and 60º by mere folding.
Let ́s go on with fig. 1. If we make IC = 1, we ́ll have:
DC = tg 3α = a
Therefore (fig. 2) we are able to draw the co-ordinate plane with origin at I (initial
point), abscissas measured in conventional units and ordinates measured with a as unit.
Looking at Fig.2 we have:
- First vector will be [I → (1,0)] (which gives the axle 2,Y)
- The second vector will be [(1,0) → (1,3a)]
- The third is [(1,3a) → (4,3a)]
As
α
α
α 2
1 tg
2 tg
tg 2
−
= , it is:
()
α
α α
α α α 2
3
1 3 tg
3 tg tg
tg 3 tg 2
−
−
= + =
Making tgα=t, we ́ll have:
2
3
1 3
3
tg 3
t
t t
−
−
α= which leads to the complete equation of third
degree:
t^3 − 3 tg 3 α×t^2 − 3 t+tg 3 α= 0 (1)
Making a=tg 3 α, (1) takes the form:
t^3 − 3 at^2 − 3 t+a= 0 (2)
B
1
I C A
2345
a
2a
3a
4a
D
1