I love parallelograms. How about you? - Correct version
I place 2 points (in pink) inside a parallelogram, and then draw a line segment from each point to each of the 4 vertices of this parallelogram, dividing it into 9 parts.
If the four triangles in the diagram have areas 13, 15, 18, and 19, respectively, what is the area of the blue quadrilateral?
The answer is 9.
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1 solution
In my solution the two points are F and H .
If we draw parallel to the parallelogram's sides through H, we make four parallelograms, and since a diagonal divides a parallelogram to two equal parts,
[
A
B
H
]
+
[
C
D
H
]
=
[
B
C
H
]
+
[
A
D
H
]
Similarly,
[
A
B
F
]
+
[
C
D
F
]
=
[
B
C
F
]
+
[
A
D
F
]
From this two equation we get
[
A
H
D
]
+
[
A
F
D
]
+
[
B
F
C
]
+
[
B
H
C
]
−
(
[
A
E
H
]
+
[
G
D
H
]
+
[
B
E
F
]
+
[
F
C
G
]
)
=
[
A
B
H
]
+
[
C
D
H
]
+
[
A
B
F
]
+
[
C
D
F
]
−
(
[
A
E
H
]
+
[
G
D
H
]
+
[
B
E
F
]
+
[
F
C
G
]
)
⇒
[
A
H
D
]
+
[
B
F
C
]
+
[
E
F
G
H
]
=
[
A
B
E
]
+
[
C
D
G
]
⇒
[
E
F
G
H
]
=
[
A
B
E
]
+
[
C
D
G
]
−
(
[
B
F
C
]
+
[
A
H
D
]
)
⇒
[
blue quadrilateral
]
=
[
green triangle
]
+
[
red triangle
]
−
(
[
yellow triangle
]
+
[
purple triangle
]
)
=
3
7
−
2
8
=
9