You could find the three vertices A, B, C of the triangle and use these to rewrite the extended lines which form the triangle boundary as
[x,y] = A + a(B-A)
for one line. Then find the intersection of that with the 4th lline. If a lies outside the unit interval, it wont pass through that side. So do it for all 3 sides.
Alternatively, you could show that all 3 vertices lie on the same side (in the same half space) of the 4th line (or not). Its probably the easiest way.
[x,y] = A + a(B-A)
for one line. Then find the intersection of that with the 4th lline. If a lies outside the unit interval, it wont pass through that side. So do it for all 3 sides.
Alternatively, you could show that all 3 vertices lie on the same side (in the same half space) of the 4th line (or not). Its probably the easiest way.
(edited 3 months ago)
mqb2766
Alternatively, you could show that all 3 vertices lie on the same side (in the same half space) of the 4th line (or not). Its probably the easiest way.
Original post by DFranklin
Given the other question the OP asked (about convexity), I'm pretty sure this would be the expected approach. There are a lot of tie-ins between convexity and half-planes (and obviously a triangle is convex).
Agreed. 10 marks seems a lot for subbing 3 points into a line inequality.
Original post by mqb2766
Agreed. 10 marks seems a lot for subbing 3 points into a line inequality.
I'm a bit confused, how does subbing the vertices shows it lie on the same side?
Original post by Amamiya
I'm a bit confused, how does subbing the vertices shows it lie on the same side?
Given a line ax + by = c, all points satisfying ax+by < c lie on one side of the line and points satisfying ax+by > c lie on the other.
[This also generalises to higher dimensions; in 3D, given the plane ax+by+cz = d, all points satisfying ax+by+cz < d lie on one side of the plane and those satisfying ax+by+cz > d lie on the other.
Same in higher dimensions (the general term is hyperplane). If you're doing all this in a course about convexity, you may well encounter https://en.wikipedia.org/wiki/Hyperplane_separation_theorem later on].
(edited 3 months ago)
Original post by Amamiya
I'm a bit confused, how does subbing the vertices shows it lie on the same side?
Just a small addition to DFranklins info. If you had two vertices which lay on either side of the line, then the line segment joining the two vertices (a triangle edge) would intersect with the line. In a sense youre determining whether theres an intersection point (or not) without actually calculating it.
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