sailorlyts14eh

2022-09-04

Let ${P}_{1}$ and ${P}_{2}$ be two convex quadrilaterals such that ${P}_{1}\ne {P}_{2}$ and $Area({P}_{1})\ge Area({P}_{2})$. Is it true that it is not possible that all sides and diagonals of ${P}_{1}$ are shorter than the corresponding sides and diagonals of ${P}_{2}$?

Suppose that all four sides and one diagonal of ${P}_{1}$ are shorter than the corresponding sides and diagonal of ${P}_{2}$. Then, as it is explained in the answer to my older question, the two triangles which triangulate ${P}_{2}$ should be "flatter" (otherwise it's not possible $Area({P}_{1})\ge Area({P}_{2})$). From this it should follow that the other diagonal of ${P}_{1}$ is longer than the corresponding diagonal of ${P}_{2}$.

Do you think it's correct? Can you help me formalizing it?

Suppose that all four sides and one diagonal of ${P}_{1}$ are shorter than the corresponding sides and diagonal of ${P}_{2}$. Then, as it is explained in the answer to my older question, the two triangles which triangulate ${P}_{2}$ should be "flatter" (otherwise it's not possible $Area({P}_{1})\ge Area({P}_{2})$). From this it should follow that the other diagonal of ${P}_{1}$ is longer than the corresponding diagonal of ${P}_{2}$.

Do you think it's correct? Can you help me formalizing it?

Domenigmh

Beginner2022-09-05Added 7 answers

It is possible that $Area({P}_{1})\ge Area({P}_{2})$ while each side and each diagonal of ${P}_{1}$ is strictly smaller than the corresponding one in ${P}_{2}$.

Take for example ${P}_{1}$ to be a square of side $1$. Take ${P}_{2}$ to be an isosceles trapezoid with the small base and the non-parallel sides all equal to 2$2$. Flatten the trapezoid while keeping those three sides at the same length $1$. In the limit the large base tends to $3\cdot 2=6$, the diagonals tend to $2\cdot 2=4\phantom{\rule{thinmathspace}{0ex}}$, and the area tends to $0\phantom{\rule{thinmathspace}{0ex}}$, yet all the sides and diagonals of ${P}_{2}$ are strictly larger than those of ${P}_{1}$. Even stronger, any side or diagonal of ${P}_{2}$ is strictly larger than any other side or diagonal of ${P}_{1}$.

Take for example ${P}_{1}$ to be a square of side $1$. Take ${P}_{2}$ to be an isosceles trapezoid with the small base and the non-parallel sides all equal to 2$2$. Flatten the trapezoid while keeping those three sides at the same length $1$. In the limit the large base tends to $3\cdot 2=6$, the diagonals tend to $2\cdot 2=4\phantom{\rule{thinmathspace}{0ex}}$, and the area tends to $0\phantom{\rule{thinmathspace}{0ex}}$, yet all the sides and diagonals of ${P}_{2}$ are strictly larger than those of ${P}_{1}$. Even stronger, any side or diagonal of ${P}_{2}$ is strictly larger than any other side or diagonal of ${P}_{1}$.

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