West Bengal Board Solutions for Chapter: Geometrical Proofs, Exercise 3: Let's work out - 20.1

$O$ is any point inside $∆ABC$. Prove that $AB+BC+AC>OA+OB+OC$.

Prove that the perimeter of a quadrilateral is more than twice the length of any diagonal.

If $P$ is any point inside the $∆ABC$, prove that $AP+PB>AB$.

If $P$ is any point inside the $∆ABC$, prove that $AB+BC+AC<2\left(AP+BP+CP\right)$.

Prove that the perimeter of a triangle is more than the sum of the lengths of the medians.

Prove that the sum of the lengths of two diagonals is greater than the sum of the lengths of any two opposite sides of a quadrilateral.

Prove that the sum of the lengths of two diagonals of a quadrilateral is more than the semi-perimeter of the quadrilateral.

From a point inside a quadrilateral $($not on any diagonal$)$, we join the vertices of the quadrilateral. Prove that the sum of the lengths of these line segments is greater than the sum of the lengths of these diagonals.