# Solution to the [[Three Regular Hexagons ii]] Puzzle +-- {.image} [[ThreeRegularHexagonsII.png:pic]] > The largest of these regular hexagons has area $9$. What's the shaded area? =-- ## Solution by [[Area of a Triangle]] and [[Area of a Regular Hexagon]] +-- {.image} [[ThreeRegularHexagonsIILabelled.png:pic]] =-- Consider the diagram with the points labelled as above. The purple region can be divided into two triangles, $A H B$ and $A C B$. To find the [[area of a triangle|area]] of each, consider them both as having base $A B$. The left-hand hexagon can vary in size, but it remains anchored at point $F$, so its diagonal remains on the line through $F$ and $H$. This is parallel to $A B$, so the height of $H$ above $A B$ is the same as that of $G$ above $A B$. So the area of triangle $A H B$ is the same as that of triangle $A G B$. A similar argument on the right-hand side shows that the area of triangle $A C B$ is the same as that of triangle $A D B$. Therefore, the purple region has the same area as rectangle $A G B D$, which, from the properties of a [[regular hexagon]], is two thirds of the area of the central hexagon. Since that hexagon has area $9$, the purple region has area $6$. ## Solution by [[Invariance Principle]] and [[Area of a Regular Hexagon]] The left and right hexagons can vary in size, meaning that this problem can be solved by appealing to the [[invariance principle]]. +-- {.image} [[ThreeRegularHexagonsIIInvarianceA.png:pic]] =-- In this first configuration, the hexagons are sized so that their diagonals coincide with sides of the central hexagon, putting points $H$ and $C$ at $G$ and $D$ respectively. This means that the purple region coincides with a central rectangle of the hexagon, so its area is two thirds that of the hexagon. +-- {.image} [[ThreeRegularHexagonsIIInvarianceB.png:pic]] =-- In this second configuration, the left and right hexagons are shrunk to points. This also makes the purple region coincide with a central rectangle, but in a different orientation.