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| 1997 East-Central problem set
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## ACM East Central Region

1997 Regional Programming Contest

### Problem 1 - Polygon Puzzler

We define a *simple polygon* as an area enclosed by
endpoint-connected line segments such that no line segment intersects
another (except for adjoining segments at their endpoints). A simple
polygon can thus be defined by an ordered list of its vertices (the
endpoints of the enclosing line segments). A *planar polygon*
is a polygon whose vertices all lie in the same plane.
For this problem you are asked to compute the area of a simple planar
polygon oriented in three space. That is, although the vertices of the
polygon lie in some two-dimensional plane, the vertices are specified
in three-dimensional Cartesian coordinates.

### Input

The input will consist of an ordered sequence of coordinates for the
vertices of the polygon. Each line of the input will contain the
three-dimensional Cartesian coordinates for a single vertex in the
order *x*, *y*, *z*. The values for the
*x*, *y*, *z* components will be separated by a
single space. Input values should be considered to be double
precision floating point and may be positive or negative. The
coordinates of the final line of input input will be the same as the
coordinates on the first line of input. No polygon will have more than
1024 vertices.
### Output

The output should be the area of the polygon specified by the input
and should be rounded to the nearest 1/1000 (i.e., three places after
the decimal point should be printed).
### Sample Input

1.401117996399998e+00 1.509291958378880e-01 1.186959898555237e-01
1.918738650437130e-01 1.067473024933127e+00 9.075713530920345e-01
1.401117996399998e+00 -1.509291958378880e-01 -1.186959898555237e-01
-1.918738650437130e-01 -1.067473024933127e+00 -9.075713530920345e-01
-1.401117996399998e+00 1.509291958378880e-01 1.186959898555237e-01

### Output for the Sample Input

4.000

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Last updated December 10, 1999