newton-gauss line proof

Tekijä:user104

If four lines intersect at 6 distinct points, the midpoints of all sets of points that do not share a line are collinear

is the midpoint of

the midpoint of

, and

the midpoint of

are collinear In the diagram above we have

the midpoint of

and

the midpoint of

. From this it is evident that

and

are parallel to

and

are parallel to

and

are parallel to

. The line we choose here doesn't really matter Now we have three points on a line and three pairs of parallel lines. Let

be the intersection of

with

. Suppose that

does not pass through

, and call the intersection of

with

and

. Then we have

which simplifies to

which probably means

This is probably a special case of something, still haven't learnt any projective geometry sadly