Assume A B C on one line and A D E on another line Show

Assume A * B * C on one line, and A * D * E on another line. Show that the segment BE must meet the segment CD at a point M. Show that the interior of a triangle is nonempty.

Solution

Given that A,B,C lie on a line and A,D,E lie on a line

So, ACE can form a triangle with B on AC and D on the line AE

consider the lines BE and CD,

These two donot meet if and only if they are parallel to each other

if they are not parallel they meet each other at some point M.

Consider the line CE as the base.

line CD will be travelling towards the line AE or towards E and

line EB will be travelling towards the line AC or towards C.

So, both the lines move in opposite directions with respect to CE or both the lines EB and CD have slopes with oppposite signs with respect to CE, which says that they can never be parallel

As EB and CD are not parallel to each other they both are going to meet at some point M.