Cannot verify Pascal's theorem

Question

In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem) states that if six arbitrary points are chosen on a conic (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon.

(Source)

So I'm given the following six points, which lie on the parabola $x+2y=x^2$: $a_1=(-2,3), a_2=(-1,1), a_3=(0,0), a_4=(1,0), a_5=(4,6), a_6=(5,10)$.

So I've drawn and redrawn my picture several times, but the points of intersection $G$, $H$, and $K$ do not line up as collinear. Can someone please point out my error?

Answer 1

I drew in the hexagram that comes from using lines 12, 23, 34, 45, 56, 61. In fact, the lines do need to be extended a little outside. Note that I had to draw the $x$ axis in orange since it is the same as the line through points 3 and 4. Well, on the paper it is clearly orange, on this screen I am having trouble as red and orange appear the same

Answer 2

here is one legitimate set of labels and edges