Equation of the line through `A` and parallel to the base `BC` is
` | (x, y, 1) , (x_1, y_1, 1) , (a, b, 1) | = 0` where `(a, b)` are assumed to be co-ordinates of `D.`
Now, equating the middle point of `BD` and `AC`
`a + x_2 = x_1 + x_3 => a = x_1 - x_2 + x_3`
`b + y_2 = y_1 + y_3 => b = y_1 - y_2 + y_3`
Hence the equation of the line is
` | (x, y, 1) , (x_1, y_1, 1) , (x_1+x_3-x_2, y_3-y_2, 1-1) | = 0 => | (x, y, 1) , (x_1, y_1, 1) , (x_2, y_2, 1) | - | (x, y, 1) , (x_1, y_1, 1) , (x_3, y_3, 1) | = 0`
Equation of the line through `A` and parallel to the base `BC` is
` | (x, y, 1) , (x_1, y_1, 1) , (a, b, 1) | = 0` where `(a, b)` are assumed to be co-ordinates of `D.`
Now, equating the middle point of `BD` and `AC`
`a + x_2 = x_1 + x_3 => a = x_1 - x_2 + x_3`
`b + y_2 = y_1 + y_3 => b = y_1 - y_2 + y_3`
Hence the equation of the line is
` | (x, y, 1) , (x_1, y_1, 1) , (x_1+x_3-x_2, y_3-y_2, 1-1) | = 0 => | (x, y, 1) , (x_1, y_1, 1) , (x_2, y_2, 1) | - | (x, y, 1) , (x_1, y_1, 1) , (x_3, y_3, 1) | = 0`