There is a set of ways to define a triangle. In analytical geometry one of these ways — to set coordinates of three of its tops. These three points define a triangle unambiguously, but to complete the picture it is necessary to work out still the equations of the parties connecting tops.
Instruction
1. To you coordinates of three points are set. Let's designate them as (x1, y1), (x2, y2), (x3, y3). It is supposed that these points are tops of some triangle. The task is in working out the equations of its parties — more precisely the equations of those straight lines on which these parties lie. These equations have to have an appearance: y = k1*x + b1; y = k2*x + b2; y = k3*x + b3. Thus, you should find slopes of k1, k2, k3 and shift of b1, b2, b3.
2. Make sure that all points are various among themselves. If some two coincide, then the triangle degenerates in a piece.
3. Find the equation of the straight line passing through points (x1, y1), (x2, y2). If x1 = x2, then a required straight line is vertical also its equation x = x1. If y1 = y2, then a straight line is horizontal also its equation y = y1. Generally these coordinates will not be equal each other.
4. Substituting coordinates (x1, y1), (x2, y2) in the general equation of a straight line, you receive a system from two linear equations: k1*x1 + b1 = y1; k1*x2 + b1 = y2. Subtract one equation from another and solve the received equation concerning k1: k1 * (x2 - x1) = y2 - y1, therefore, k1 = (y2 - y1) / (x2 - x1).
5. Substituting the found expression in any of the initial equations, find expression for b1: ((y2 - y1) / (x2 - x1)) *x1 + b1 = y1; b1 = y1 - ((y2 - y1) / (x2 - x1)) *x1. As it is already known that x2 ≠ x1, it is possible to simplify expression, having increased y1 on (x2 - x1) / (x2 - x1). Then for b1 you receive the following expression: b1 = (x1*y2 - x2*y1) / (x2 - x1).
6. Check whether the third of the set points on the found straight line lies. For this purpose substitute values (x3, y3) in the removed equation and look whether equality is observed. If it is observed, therefore, all three points lie on one straight line, and the triangle degenerates in a piece.
7. In the same way that it is described above, remove the equations for the straight lines passing through points (x2, y2), (x3, y3) and (x1, y1), (x3, y3).
8. The final type of the equations for the parties of the triangle set by coordinates of tops looks so: (1) y = ((y2 - y1) *x + (x1*y2 - x2*y1)) / (x2 - x1); (2) y = ((y3 - y2) *x + (x2*y3 - x3*y2)) / (x3 - x2); (3) y = ((y3 - y1) *x + (x1*y3 - x3*y1)) / (x3 - x1).