Question 118
We call a spatial polygon (a polygon in the three dimensional space) **Latticelike** if its edges are parallel to the coordinate axes.
a) For an arbitrary Latticelike polygon, any two consecutive edges form a right angle that lies in either $xy$, $yz$ or $zx$ plane. Prove that the number of angles of these three kinds have the same parity.
b) A Latticelike polygon is called an Inscribed Latticelike polygon if there exists a point in space that has equal distances from every vertex of the polygon. Prove that if a hexagon is not planar (i.e. there doesn't exist a plane containing all six vertices), then
it's an Inscribed Latticelike polygon.
c) Does there exist a Latticelike 2014-gon with distinct vertices for which there exists a plane that intersects every edge of it in an inner point (i.e. a point other than its two vertices)?
d) $a$, $b$ and $c$ are three natural numbers greater than 1. Prove that one can find three points in plane having mutual distance equal to $a$, $b$ and $c$, if and only if there exists a Latticelike polygon with $a$, $b$ and $c$ edges in each of the three coordinate directions.
a) For an arbitrary Latticelike polygon, any two consecutive edges form a right angle that lies in either $xy$, $yz$ or $zx$ plane. Prove that the number of angles of these three kinds have the same parity.
b) A Latticelike polygon is called an Inscribed Latticelike polygon if there exists a point in space that has equal distances from every vertex of the polygon. Prove that if a hexagon is not planar (i.e. there doesn't exist a plane containing all six vertices), then
it's an Inscribed Latticelike polygon.
c) Does there exist a Latticelike 2014-gon with distinct vertices for which there exists a plane that intersects every edge of it in an inner point (i.e. a point other than its two vertices)?
d) $a$, $b$ and $c$ are three natural numbers greater than 1. Prove that one can find three points in plane having mutual distance equal to $a$, $b$ and $c$, if and only if there exists a Latticelike polygon with $a$, $b$ and $c$ edges in each of the three coordinate directions.