ax+by+c=0If b is not 0, the slope of that line is -a/b .
Take any two lines
a x + b y + c = 0 d x + e y + f = 0 The lines are parallel <=> A direction vector of the first line is a multiple of the direction vector of the second line. <=> (b, -a) is a multiple of (e, -d) <=> (a, b) is a multiple of (d, e) <=> a e = b d <=> | a b | | d e | = 0
The lines a x + b y + c = 0 and d x + e y + f = 0 are parallel <=> a e = b d <=> | a b | | d e | = 0 |
Three points P(x_{1},y_{1}) Q(x_{2},y_{2}) and R(x_{3},y_{3}) are collinear <=> There is a line a x + b y + c = 0 that contains the points P, Q and R. <=> There is a line a x + b y + c = 0 such that a x_{1} + b y_{1} + c = 0 a x_{2} + b y_{2} + c = 0 a x_{3} + b y_{3} + c = 0 <=> There is an a, b and c, not all zero, such that a x_{1} + b y_{1} + c = 0 a x_{2} + b y_{2} + c = 0 a x_{3} + b y_{3} + c = 0 <=> The following homogeneous system with unknowns a, b and c has a solution different from (0,0,0). a x_{1} + b y_{1} + c = 0 a x_{2} + b y_{2} + c = 0 a x_{3} + b y_{3} + c = 0 <=> |x_{1} y_{1} 1| |x_{2} y_{2} 1| = 0 |x_{3} y_{3} 1|Conclusion :
Three points P(x_{1},y_{1}) Q(x_{2},y_{2}) and R(x_{3},y_{3}) are collinear if and only if
|x_{1} y_{1} 1| |x_{2} y_{2} 1| = 0 |x_{3} y_{3} 1| |
|x y 1| |x1 y1 1| = 0 |x2 y2 1|So, this is the equation of the line PQ.
The line PQ with P(x_{1},y_{1}) and Q(x_{2},y_{2}) is
|x y 1| |x1 y1 1| = 0 |x2 y2 1| |
Three lines
a x + b y + c = 0 a' x + b' y + c' = 0 a" x + b" y + c" = 0are concurrent if and only if | a b c | | a' b' c'| = 0 | a" b" c"| |
a x + b y = 1 (1) a x + y = b (2) x + b y = a (3)are the equations of three lines. The parameters a and b are real and different. Examine the relative position of the three lines for all values of a and b. |
(1) and (2) are parallel if and only if ( a = 0 of b = 1 )
(1) and (3) are parallel if and only if ( b = 0 of a = 1 )
(2) and (3) are parallel if and only if ( a b = 1 )
We now treat these cases separately
The equations of the lines are b y = 1 (1) y = b (2) x + b y = 0 (3)The lines (1) and (2) are parallel and the third one intersects (1) and (2).
The equations of the lines are a x = 1 (1) a x + y = 0 (2) x = a (3)The lines (1) and (3) are parallel and the second one intersects (1) and (3).
The equations of the lines are x + b y = 1 (1) x + y = b (2) x + b y = 1 (3)The lines (1) and (3) coincides and the second one intersects (1) and (3).
The equations of the lines are a x + y = 1 (1) a x + y = 1 (2) x + y = a (3)The lines (1) and (2) coincides and the third one intersects (1) and (2).
The equations of the lines are a x + (1/a) y = 1 (1) a x + y = (1/a) (2) x + (1/a) y = a (3)The lines (2) and (3) are parallel but they don't coincide.
a and b are different from 1 and from 0 and a.b is not 1.
The three lines are concurrent if and only if
| a b 1 | | a 1 b | = 0 | 1 b a | <=> (a-1)(b-1)(a+b+1) = 0Since a and b are not equal to 1, we have :
The three lines form a triangle.
___________________________ | \| (x_{2} - x_{3})^{2} + (y_{2} - y_{3})^{2}From above, the equation of the line QR is
|x y 1| |x2 y2 1| = 0 |x3 y3 1|If we calculate this determinant emanating from the first row, we find
x(y_{2} - y_{3}) - y(x_{2}-x_{3}) + x_{2} y_{3} - x_{3} y_{2} = 0To calculate the distance from P to the line QR, we write first the normal equation of a line QR
x(y_{2} - y_{3}) - y(x_{2} - x_{3}) + x_{2} y_{3} - x_{3} y_{2} --------------------------------------- = 0 _________________________ | \| (x_{2} - x_{3})^{2} + (y_{2} - y_{3})^{2} <=> |x y 1| |x2 y2 1| |x3 y3 1| --------------------------------------- = 0 ___________________________ | \| (x_{2} - x_{3})^{2} + (y_{2} - y_{3})^{2}Now, to find the distance, we have to take the absolute value of the left side and we must replace x and y by the coordinates of P. The distance from P to QR is
|x_{1} y_{1} 1| |x_{2} y_{2} 1| |x_{3} y_{3} 1| | -------------------------------- | _________________________ | \| (x_{2} - x_{3})^{2} + (y_{2} - y_{3})^{2}The area of the triangle P(x_{1},y_{1}) Q(x_{2},y_{2}) and R(x_{3},y_{3}) is
|x_{1} y_{1} 1| _________________________ |x_{2} y_{2} 1| 1 | |x_{3} y_{3} 1| - . \| (x2 - x3)^{2} + (y2 - y3)^{2} | -------------------------------- | 2 _________________________ | \| (x_{2} - x_{3})^{2} + (y_{2} - y_{3})^{2} <=> 1 |x_{1} y_{1} 1| -.| |x_{2} y_{2} 1| | 2 |x_{3} y_{3} 1|This is a very simple formula to calculate the area of a triangle.
The area of the triangle P(x_{1},y_{1}) Q(x_{2},y_{2}) and R(x_{3},y_{3}) is
1 |x_{1} y_{1} 1| -.| |x_{2} y_{2} 1| | 2 |x_{3} y_{3} 1| |