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Question
ABCD is a parallelogram with vertices \[A ( x_1 , y_1 ), B \left( x_2 , y_2 \right), C ( x_3 , y_3 )\] . Find the coordinates of the fourth vertex D in terms of \[x_1 , x_2 , x_3 , y_1 , y_2 \text{ and } y_3\]
Solution
Suppose the coordinates of D be (x, y).
Since diagonals of a parallelogram bisect each other.
Therefore the midpoint of AC is the midpoint of BD, i.e
\[\Rightarrow x_1 + x_3 = x_2 + x \text{ and } y_1 + y_3 = y_2 + y\]
\[ \Rightarrow x = x_1 + x_3 - x_2 \text{ and } y = y_1 + y_3 - y_2 \]
\[\text{ Thus coordinates of D are } \left( x_1 + x_3 - x_2 , y_1 + y_3 - y_2 \right)\]
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