Points A, B, C, D are lying on a square of integer side length such that each point divides the corresponding side into integer segments.
If AC=2√26 and BD=2√29, what is the maximum area of the quadrilateral ABCD?
For side = 5 To 30
side2 = side * side
For a = 0 To side
For c = 0 To side
ac2 = (c - a) * (c - a) + side2
If ac2 = 104 Then
For b = 0 To side
For d = 0 To side
bd2 = (d - b) * (d - b) + side2
If bd2 = 116 Then
area = side * side - a * d / 2 - (side - a) * b / 2 - (side - b) * (side - c) / 2 - c * (side - d) / 2
Text1.Text = Text1.Text & a & Str(b) & Str(c) & Str(d) & " " & side & Str(area) & crlf
If area > maxarea Then maxarea = area
End If
Next
Next
End If
Next
Next
Next
assumes a is on the opposite side from c and finds
where
a is A's position to the right of the top-left
b is B's position below the top-right
c is C's position to the right of the bottom left
d is D's position below the top-left
square
side quadrilateral
a b c d length area
3 10 1 6 10 54
3 0 5 4 10 54
3 1 5 5 10 54
3 2 5 6 10 54
3 3 5 7 10 54
3 4 5 0 10 46
3 4 5 8 10 54
3 5 5 1 10 46
3 5 5 9 10 54
3 6 5 2 10 46
3 6 5 10 10 54
3 7 5 3 10 46
3 8 5 4 10 46
3 9 5 5 10 46
3 10 5 6 10 46
4 0 2 4 10 46
4 1 2 5 10 46
4 2 2 6 10 46
4 3 2 7 10 46
4 4 2 0 10 54
4 4 2 8 10 46
4 5 2 1 10 54
4 5 2 9 10 46
4 6 2 2 10 54
4 6 2 10 10 46
4 7 2 3 10 54
4 8 2 4 10 54
4 9 2 5 10 54
4 10 2 6 10 54
4 0 6 4 10 54
4 1 6 5 10 54
4 2 6 6 10 54
4 3 6 7 10 54
4 4 6 0 10 46
4 4 6 8 10 54
4 5 6 1 10 46
4 5 6 9 10 54
4 6 6 2 10 46
4 6 6 10 10 54
4 7 6 3 10 46
4 8 6 4 10 46
4 9 6 5 10 46
4 10 6 6 10 46
5 0 3 4 10 46
5 1 3 5 10 46
5 2 3 6 10 46
5 3 3 7 10 46
5 4 3 0 10 54
5 4 3 8 10 46
5 5 3 1 10 54
5 5 3 9 10 46
5 6 3 2 10 54
5 6 3 10 10 46
5 7 3 3 10 54
5 8 3 4 10 54
5 9 3 5 10 54
5 10 3 6 10 54
5 0 7 4 10 54
5 1 7 5 10 54
5 2 7 6 10 54
5 3 7 7 10 54
5 4 7 0 10 46
5 4 7 8 10 54
5 5 7 1 10 46
5 5 7 9 10 54
5 6 7 2 10 46
5 6 7 10 10 54
5 7 7 3 10 46
5 8 7 4 10 46
5 9 7 5 10 46
5 10 7 6 10 46
6 0 4 4 10 46
6 1 4 5 10 46
6 2 4 6 10 46
6 3 4 7 10 46
6 4 4 0 10 54
6 4 4 8 10 46
6 5 4 1 10 54
6 5 4 9 10 46
6 6 4 2 10 54
6 6 4 10 10 46
6 7 4 3 10 54
6 8 4 4 10 54
6 9 4 5 10 54
6 10 4 6 10 54
6 0 8 4 10 54
6 1 8 5 10 54
6 2 8 6 10 54
6 3 8 7 10 54
6 4 8 0 10 46
6 4 8 8 10 54
6 5 8 1 10 46
6 5 8 9 10 54
6 6 8 2 10 46
6 6 8 10 10 54
6 7 8 3 10 46
6 8 8 4 10 46
6 9 8 5 10 46
6 10 8 6 10 46
7 0 5 4 10 46
7 1 5 5 10 46
7 2 5 6 10 46
7 3 5 7 10 46
7 4 5 0 10 54
7 4 5 8 10 46
7 5 5 1 10 54
7 5 5 9 10 46
7 6 5 2 10 54
7 6 5 10 10 46
7 7 5 3 10 54
7 8 5 4 10 54
7 9 5 5 10 54
7 10 5 6 10 54
7 0 9 4 10 54
7 1 9 5 10 54
7 2 9 6 10 54
7 3 9 7 10 54
7 4 9 0 10 46
7 4 9 8 10 54
7 5 9 1 10 46
7 5 9 9 10 54
7 6 9 2 10 46
7 6 9 10 10 54
7 7 9 3 10 46
7 8 9 4 10 46
7 9 9 5 10 46
7 10 9 6 10 46
8 0 6 4 10 46
8 1 6 5 10 46
8 2 6 6 10 46
8 3 6 7 10 46
8 4 6 0 10 54
8 4 6 8 10 46
8 5 6 1 10 54
8 5 6 9 10 46
8 6 6 2 10 54
8 6 6 10 10 46
8 7 6 3 10 54
8 8 6 4 10 54
8 9 6 5 10 54
8 10 6 6 10 54
8 0 10 4 10 54
8 1 10 5 10 54
8 2 10 6 10 54
8 3 10 7 10 54
8 4 10 0 10 46
8 4 10 8 10 54
8 5 10 1 10 46
8 5 10 9 10 54
8 6 10 2 10 46
8 6 10 10 10 54
8 7 10 3 10 46
8 8 10 4 10 46
8 9 10 5 10 46
8 10 10 6 10 46
9 0 7 4 10 46
9 1 7 5 10 46
9 2 7 6 10 46
9 3 7 7 10 46
9 4 7 0 10 54
9 4 7 8 10 46
9 5 7 1 10 54
9 5 7 9 10 46
9 6 7 2 10 54
9 6 7 10 10 46
9 7 7 3 10 54
9 8 7 4 10 54
9 9 7 5 10 54
9 10 7 6 10 54
10 0 8 4 10 46
10 1 8 5 10 46
10 2 8 6 10 46
10 3 8 7 10 46
10 4 8 0 10 54
10 4 8 8 10 46
10 5 8 1 10 54
10 5 8 9 10 46
10 6 8 2 10 54
10 6 8 10 10 46
10 7 8 3 10 54
10 8 8 4 10 54
10 9 8 5 10 54
10 10 8 6 10 54
The maximum area of ABCD is 54 for example A, B, C, D at 6, 5, 4, 1 from respective side ends on a 10x10 square:
+ + + + + + A + + + +
D + The cutoff triangles have area
+ + 3, 10, 15 and 18, starting at AD
+ + and going clockwise.
+ +
+ B
+ +
+ +
+ +
+ +
+ + + + C + + + + + +
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Posted by Charlie
on 2019-05-04 13:53:04 |
computer solution
