Triangles?
True or false :
A triangle with integer coordinates on a Cartesian plane will have area a such that 2 a is always an integer.
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2 solutions
Let the vertices be
A
(
x
1
,
y
1
)
,
B
(
x
2
,
y
2
)
and
C
(
x
3
,
y
3
)
where
x
1
,
x
2
,
x
3
,
y
1
,
y
2
,
y
3
are integers.
Area of
△
A
B
C
(
a
)
=
2
1
∣
x
1
(
x
2
−
x
3
)
+
x
2
(
x
3
−
x
1
)
+
x
3
(
x
1
−
x
2
)
∣
.
Since
∣
x
1
(
x
2
−
x
3
)
+
x
2
(
x
3
−
x
1
)
+
x
3
(
x
1
−
x
2
)
∣
leads to an integer(say
m
)
,
a
=
2
1
(
m
)
.
Therefore,
2
a
=
m
where
m
is an integer.
Any such triangle can be surrounded by a rectangle and the area of the triangle then calculated as the area of the tightly fitting rectangle minus the areas of three right triangles outside of it, each of which has integer sides. The area of the rectangle, since it too has integer sides, is an integer. The areas of the triangles are integers divided by 2. Therefore if their difference is multiplied by 2, it too becomes an integer.