A Problem With A Happy Ending
By Paulo E. D. Pinto, UERJ- Universidade do Estado do Rio de Janeiro 
Brazil
By Paulo E. D. Pinto, UERJ- Universidade do Estado do Rio de Janeiro Brazil
Paul Erdős was a noted Hungarian mathematician who published 1475 articles in several areas. He gave the name of "Theorem of the Happy End" to the following theorem proved by him and some friends:
"Given 5 points in the plan, not collinear three to three, four of them always form a convex quadrilateral. "
The name of the theorem is due to the fact that one couple of friends who worked on it ended up getting married. Many mathematicians worked on the generalization of this problem, which still was not achieved. In this problem you will prove the theorem and, hopefully, with total happiness! Are given 5 points, not collinear three to three, and you should find the area of the largest convex quadrilateral involving 4 of the given points.
Each input line contain 10 even integers X1, Y1, X2, Y2, X3, Y3, X4, Y4, X5, Y5, 0 ≤ Xi, Yi ≤ 103, 1 ≤ i ≤ 5, where each pair Xi, Yi represents the coordinates of each one of the five points. The input ends with 10 values 0 that should not be processed.
For each input, print the area of the biggest convex quadrilateral that can be formed with 4 of the 5 points given.
| Input Sample | Output Sample |
|
0 100 0 0 100 100 50 90 100 0 |
10000 |