From a point insi

In the figure shown below, ABCD is a quadrilateral and AC and BD are the two diagonals of quadrilateral.

Also,

E is a point inside the quadrilateral,

We need to prove that,

AE + BE + ED + EC > AC + BD

Now, we know that the straight line is the shortest distance between two points. Therefore, AC will be the shortest distance between A and C and thus AC > AE + EC….(1)

Also,

BD > BE + ED….(2)

Adding (1) and (2), we get,

AC + BD > AE + EC + BE + ED

Hence, Proved.

We can see that at the point where both the diagonals meet the lengths of the line segment obtained by the vertices of the quadrilateral with that point will be smallest.