The points (0,0), (a,0) and (B,C) are the vertices of an arbitrar triangle which is placed in a coninesvenient position relative to the coordinate system.

a) find the equation of the line through each vertex perpendicular to the opposite side, and show algebraically that these 3 lines intersect at a single point.

b) find the equation of the perpendicular bisector of each side, and show algebraically that these 3 lines intersect at a single point. Why is this fact geometrically obvious?

c) find the equation of the line through each vertex and the midpoint of the opposite side, and show algebraically that these 3 lines intersect at a single point. Also verif that this point is 2/3rds of the way from each vertex to the midpoint of the opposite side

Im not really good at answering questions like this. Im better at like, just solving an equation with numbers in my face. How do i show this algebraicly? like get the slopes of each line and find a way to substitute them into each other for the point somehow?