neighbour in maximal planar graph

Question

Suppose we have a maximal planar graph G. Fix three vertices $x,y,z \in V$ such that these vertices form a triangle in G.

Suppose $\deg x >2$. My question is the following:

Does there always exists $u,v \in Adj(x)$ such that:

1. $(u,y) \in E$
2. $(v,z) \in E$

Here $u$ and $v$ need not be different. (For example in $K_4$, $u=v$)

I've tried showing this but with no luck. I have a feeling that if 1. and 2. are not true then we could add the edges while keeping the graph planar giving a contradiction.

Any hints/explanations/counter-examples would be appreciated.