If in a parallelogram ABDC , the coordinates of A , B and C are respectively (1,2),(3,4) and ( 2 , 5 ) , then the equation of the diagonal AD is :

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Since, in parallelogram mid points of both diagonals coinsides. $∴$ mid-point of $A D =$ mid-point of $BC$
$(\frac{x _{1} + 1}{2}, \frac{y _{1} + 2}{2}) = (\frac{3 + 2}{2}, \frac{4 + 5}{2})$ $∴ (x_{1}, y_{1}) = (4, 7)$ Then, equation of $A D$ is, $y - 7 = \frac{2 - 7}{1 - 4} (x - 4)$ $y - 7 = \frac{5}{3} (x - 4)$ $3 y - 21 = 5 x - 20$ $5 x - 3 y + 1 = 0$

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