Equation of the median through `A(x_1, y_1)` is

`| (x, y, 1) , (x_1, y_1, 1) , ((x_2+x_3)/2 , (y_2+y_3)/2, 1) | = 0` or `| (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 internal and external angle bisectors of `A` are

`b | (x, y, 1) , (x_1, y_1, 1) , (x_2, y_2, 1) | pm c | (x, y, 1) , (x_1, y_1, 1) , (x_3, y_3, 1) | = 0`

Equation of the altitude through `'A'` is

`b cos C | (x, y, 1) , (x_1, y_1, 1) , (x_2, y_2, 1) | + c cos B | (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`