Generating Polygons with Triangles - Discrete and Computational Geometry and Graphs - page 118

Proof. (i) By Corollary 3.1 and Theorem 3.3, it follows that among the triangles
with commensurable angles, ( 6 , 6 , 2 3 ) is only one intermediate triangle. (ii)

( 6 , 6 , 2 3 ) and suppose that ˃

Let ˃ = ˄ . Then the angles of ˄ are

also commensurable and ˄ is not a right triangle by Theorem 3.3(i). Since ˃ is
an intermediate triangle, and ˃

ₒ ˄,˃ ∼

˄ , it follows from Theorem 3.1(i)(ii) that ˄
is an equilateral triangle, which is terminal. Suppose now η

∼

ₒ ˃, η ∼ ˃ . Then

( 6 , 3 , 2 ), which is original by Theorem 3.3.

by Theorem 3.1(ii), we have η =

( 6 , 6 , 2 3 ).

Hence ( ∗ ) is a unique generating chain of length ≥ 3 that contains

Let ˃ be a triangle with non-commensurable angles, and suppose that ˃ ₒ ˄

and ˃ ˄ . From Laczkovich [ 7 , Theorems 4.1], all possible candidates for such

pair ( ˃, ˄ ) are derived as in Table 1 .
The next theorem follows from Table 1 .

∼

Table 1. Possible pairs ( ˃, ˄ ) when angles of ˃ are non-commensurable

˃ ˄

( ʱ, ˀ− 2 ʱ
2

, 2 ) 1. ( ʱ, ʱ, ˀ − 2 ʱ )

ʱ, 3 , 2 ˀ− 3 ʱ ( 3 , 3 , 3 2. ( ) )

3

3. ( ʱ, 2 ʱ, ˀ − 3 ʱ ) ( ʱ, ʱ, ˀ − 2 ʱ )

( ʱ, ˀ− 3 ʱ
2

, ˀ + ʱ
2

4. ) ( ʱ, ʱ, ˀ − 2 ʱ )or

(3 ʱ, ˀ− 3 ʱ
2

, ˀ− 3 ʱ
2

)or
( ʱ, 2 ʱ, ˀ − 3 ʱ )or
( ʱ, ˀ− 2 , ˀ−ʱ

)or

2

ʱ, ˀ− 2 , ˀ− 3 ʱ

(2 )

2

( ʱ, ˀ− 3 ʱ
3

, 2 3 ) 5. ( ʱ, ʱ, ˀ − 2 ʱ )or

ʱ, ʱ, ˀ − ʱ

( 2 3 )or

( ʱ, ˀ +3 ʱ
3

, 2 ˀ− 6 ʱ
3

)or

( ʱ, 3 , 2 ˀ− 3 ʱ )or

(2 ʱ, 3 , 2 ˀ− 6 ʱ

3

)or

3

( 3 , 3 , 3 )

Theorem 3.4. If ˃ is a triangle with non-commensurable angles, then all pos-
sible types of generating chains η ₒ ˃ ₒ ˄ of length three are given as follows:

( ʱ, ˀ− 3 ʱ
2

, ˀ + ʱ
2

( a ) ) ₒ ( ʱ, 2 ʱ, ˀ − 3 ʱ ) ₒ ( ʱ, ʱ, ˀ − 2 ʱ )

( ʱ, ˀ− 3 ʱ
3

, 2 3 )

( b ) ₒ ( ʱ, 2 ʱ, ˀ − 3 ʱ ) ₒ ( ʱ, ʱ, ˀ − 2 ʱ )

( ʱ, ˀ− 3 ʱ
3

, 2 3 ) ( ʱ, 3 , 2 ˀ− 3 ʱ ( 3 , 3 , 3 ) ( c ) ₒ ) ₒ

3

( ʱ, ˀ− 3 ʱ
3

, 2 3 ) (2 ʱ, 3 , 2 ˀ− 6 ʱ ( 3 , 3 , 3 ) .

( d ) ₒ ) ₒ

3