# Solution to the Two Regular Hexagons (II) Puzzle +-- {.image} [[TwoRegularHexagonsII.png:pic]] > Both hexagons are regular. How long is the pink line? =-- ## Solution by [[Lengths in a Regular Hexagon]] and [[Similar Triangles]] +-- {.image} [[TwoRegularHexagonsIILabelled.png:pic]] =-- Consider the triangles $A B C$ and $F H I$ in the diagram. The length of $A B$ is half the length of $F H$. The length of $I H$ is the difference of the sides. Using [[lengths in a regular hexagon]], $A D$ and $C E$ have lengths half the side lengths of their respective hexagons, so the length of $C B$ is half the difference of these sides and so has half the length of $I H$. Since both $A B C$ and $F H I$ are [[right-angled triangles]], this establishes them as [[similar]] with a scale factor of $2$. Therefore the length of $F I$ is twice that of $A C$ and so is of length $12$. ## Solution by [[Invariance Principle]] +-- {.image} [[TwoRegularHexagonsIISame.png:pic]] =-- By drawing the hexagons the same size, the relationship between the two lengths is clear and so the required length is $12$.