Making a Strip of Isosceles Trapezoids

picture

construction

ϵ

-0.3

m

1

Suppose the angle

D

1

A

1

B

1

measures

2π/5+ϵ

. The line

A

1

C

1

bisects

∠D

1

A

1

B

1

, so

|∠

D

1

A

1

C

1

|=π/5+ϵ/2

. The line

C

1

D

1

bisects

∠

A

1

C

1

D

1

, so triangle

A

1

D

1

C

1

is isosceles and

∠A

1

D

1

C

1

is congruent to

∠B

2

C

1

D

1

, which measures

2π/5-ϵ/4

. Then

|

∠B

2

C

1

A

2

|=π/5-ϵ/8

and

|∠

D

2

A

2

B

2

|=1/2(π-(π/5-ϵ/8))=2π/5+ϵ/16

. So the angles

D

n

A

n

B

n

converge to

2π/5

. In this way the trapezoids

A

n

D

n

C

n

D

n

converge to an isosceles trapezoid congruent to the trapezoid consisting of three sides and of a diagonal of a regular pentagon. Isosceles trapezoids are obtained immediately when

ϵ=0

. Otherwise, the isosceles condition is approached as the number of repeats,

m∞

. Visually, however, the result appears to be satisfied with

m≈5

. Use the bottom scroll bar to view extended regions of the diagram.