We show that if $BD+BE=2CH$, then $BXYC$ is cyclic. Observe that if $D', E'$ are the reflections of $D, E$ with respect to $B, C$, the length condition rewrites as $HD'=HE'$, so $AD'=AE'$ and, moreover, $ACXD'$ and $ABYE'$ are cyclic. We will show that $DEXY$ is cyclic. Firstly, notice that this finishes the problem: if $XY \cap BC=Q$ and $P$ is the Miquel point of $DEXY$, then $P \in (ABC)$ by spiral similarity and it is well-known that $P \in AQ$. Thus, $$QX \cdot QY=QD \cdot QE=QP \cdot QA=QB \cdot QC, $$so $BXYC$ is cyclic. Now, see the image below for a proof that $DEXY$ is cyclic.

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