Question

Using coordinate geometry to prove that three altitudes are concurrent

Answer 1

Answer:

Step-by-step explanation:

If we assume that two vertices of the triangle lie on the x-axis (which can be obtained by rotation and translation) then the vertices are (0,0), (x1,0) and (x2,y2). Let us try to find the equations of the altitudes.

The first one is easy

x=x2(1).

What about second one? We want to find a line with normal vector (x2−x1,y2) which goes trough the point (0,0). (We know the normal vector since the line has to be to perpendicular on the segment connecting the points (x1,0) and (x2,y2).) Using standard form of the equation of line we get

(x2−x1)x+y2y=0(2).

The third of them: We are looking for a line with normal vector (x2,y2) which goes through the point (x1,0). This gives the equation

x2x+y2y=x2x1(3).

The only one thing is missing - to show that the system of linear equations (1), (2) and (3) has a solution.