Notes
three adjacent triangles solution

Solution to the Three Adjacent Triangles Puzzle

Solution by Area of a Triangle

Consider the diagram with the points labelled as above. Starting on the left, triangles $A B J$ and $B C J$ have the same area and the same height above the line $A C$, so the lengths of $A B$ and $B C$ must be the same. As $A C$ has length $30$, $B C$ therefore has length $15$.

On the right, triangle $E F G$ has the same height as the full triangle $C F G$ but a fifth of the area, so $E F$ must be one fifth of $C F$, which is $6$, meaning that $C E$ is the remaining $24$. Similarly, triangle $D E H$ is one third of triangle $C E H$ so $D E$ is one third of $C E$, which is $8$, meaning that $C D$ is the remaining $16$.

Putting these together, $B D$ has length $31$.