Crossword-Solution: COLLINEAR 9 letters, 4 clues 🏆 scrabble score: 11

Extend problem 4 as follows: If a variable polygon of an even number of sides move in such a way as always to remain inscribed in a fixed conic, while all its sides but one pass through as many fixed collinear points, then the last side will also pass through a fixed point collinear with the others.

The triangles are "co-axial" in virtue of the property that the meets of corresponding sides are collinear and copolar, since the lines joining corresponding vertices are concurrent.

From axioms 1-5 it can be proved that any two distinct points in a straight line determine that line, that any three non-collinear points in a plane determine that plane, that the straight line containing any two points in a plane lies wholly in that plane, and that any two straight lines in a plane intersect.

The enunciation of this theorem is as follows: If ABC and A'B'C' are two coplanar triangles such that the lines AA', BB', CC' are concurrent, then the three points of intersection of BC and B'C' of CA and C'A', and of AB and A'B' are collinear; and conversely if the three points of intersection are collinear, the three lines are concurrent.

The couple B and D is said to separate A and C, if the four points are collinear and D lies in the segment complementary to the segment ABC.