Notes
chords dividing a triangle into five solution

Solution to the Chords Dividing a Triangle into Five Puzzle

Solution by the Intersecting Chords Theorem

With the points labelled as above, the intersecting chords theorem shows that triangles $C B D$ and $C E A$ are similar, with $D$ corresponding to $A$ and $B$ to $E$. It also shows that triangles $F D B$ and $F A E$ are also similar, also with $D$ corresponding to $A$ and $B$ to $E$. Since both pairs of similar triangles share the sides $B D$ and $E A$, and for both pairs these sides are corresponding, the scale factor from $C B D$ to $C E A$ is the same as that from $F D B$ to $F A E$. The area scale factor from $F D B$ to $F A E$ is $5$, so this is also the area scale factor from $C B D$ to $C E A$. The total area of the triangle is thus $35$.