Question 95
Suppose $ABCD$ is a simple quadrilateral, with side lengths
$$
a = |BC|,\ b = |CD|,\ c = |DA|,\ d = |AB|.
$$
Let $p = |AC|$. Prove that
$$
\max(|d-a|,\ |b-c|) \lt p \lt \min(d+a,\ b+c).
$$
Conversely, if five positive numbers $a, b, c, d, p$ satisfy this condition, prove that $a, b, c, d$ are the side lengths of a simple quadrilateral, and that $p$ is the length of a diagonal.
$$
a = |BC|,\ b = |CD|,\ c = |DA|,\ d = |AB|.
$$
Let $p = |AC|$. Prove that
$$
\max(|d-a|,\ |b-c|) \lt p \lt \min(d+a,\ b+c).
$$
Conversely, if five positive numbers $a, b, c, d, p$ satisfy this condition, prove that $a, b, c, d$ are the side lengths of a simple quadrilateral, and that $p$ is the length of a diagonal.