For any $n \geq 4$, I was able to prove that every Apollonian network has $3n - 8$ triangles. An Apollonian network is a planar graph defined by recursively subdividing a triangle by three smaller triangles (adding a new vertex inside the face and connecting it to all three vertices). Now I am trying to prove a similar statement for arbitrary planar graphs. In particular, I am trying to show that every planar graph has at most $an + b$ triangles, for some constants $a$ and $b$, where the goal is to make the constants as tight as possible. The result I showed implies that we need $a \geq 3$ and $b \geq -8$ This is an assignment question, so I would only like a hint please. I believe that I should be able to prove that no planar graph has more triangles than an Apollonian network, but unsure how to proceed.