Let $p,q$ be two projections of a $C^*$ algebra. A projection $l$ is called a bisector projection to $(p,q)$ if $$|pl-l|=|ql-l|$$ The motivation comes from the geometric intuition of "Bisector" in plane geometry.
Assume that two projections $p,q$ are similar or Mourray von Neumann equivalent. Does they admit a bisector projection?
It can be easily shown that every two homotopic projections $p,q$ have a bisector projection. Here is a proof: Let $\gamma (t)$ be a curve of projections with $\gamma(0)=p,\; \gamma(1)=q$. Now we apply the intermediate value theorem to the continuous function $$\phi (t)= |p\gamma(t)-\gamma(t)|-|q\gamma (t)-\gamma(t)|$$ In fact we have $\phi(0)\phi(1)<0$. For if $pq=q$ then $qp=q$ then $e=p-q$ would be an idempotent with $e^*=-e$, a contradiction.