WOLFRAM|DEMONSTRATIONS PROJECT

Hoehn's Theorem

​
V
1
W
1
≈
10.07
V
1
W
2
≈
13.79
V
1
W
5
≈
9.79
V
1
W
4
≈
13.58
V
2
W
1
≈
10.07
V
2
W
5
≈
14.27
V
2
W
2
≈
7.95
V
2
W
3
≈
12.93
V
3
W
2
≈
6.26
V
3
W
1
≈
9.98
V
3
W
3
≈
7.56
V
3
W
4
≈
11.71
V
4
W
3
≈
5.30
V
4
W
2
≈
10.28
V
4
W
4
≈
4.51
V
4
W
5
≈
8.30
V
5
W
4
≈
7.50
V
5
W
3
≈
11.65
V
5
W
5
≈
7.50
V
5
W
1
≈
13.38
V
1
W
1
V
3
W
2
×
V
2
W
2
V
4
W
3
×
V
3
W
3
V
5
W
4
×
V
4
W
4
V
1
W
5
×
V
5
W
5
V
2
W
1
=
1
V
1
W
2
V
3
W
1
×
V
2
W
2
V
4
W
2
×
V
3
W
4
V
5
W
3
×
V
4
W
5
V
1
W
4
×
V
5
W
1
V
2
W
5
=
1
Draw the diagonals in a pentagon with vertices
V
1
,
V
2
,
V
3
,
V
4
, and
V
5
. Let the points of intersection of the diagonals be
W
1
,
W
2
,
W
3
,
W
4
, and
W
5
. Then the following hold:
​
​
V
1
W
1
V
3
W
2
×
V
2
W
2
V
4
W
3
×
V
3
W
3
V
5
W
4
×
V
4
W
4
V
1
W
5
×
V
5
W
5
V
2
W
1
=1
,
V
1
W
2
V
3
W
1
×
V
2
W
2
V
4
W
2
×
V
3
W
4
V
5
W
4
×
V
4
W
5
V
1
W
4
×
V
5
W
1
V
2
W
5
=1
.