My first attempt:
\[CapitalDelta]reg =
DiscretizeRegion[Triangle[], MaxCellMeasure -> 1/10]
polys = MeshPrimitives[\[CapitalDelta]reg, 2]
mcells = MeshCells[\[CapitalDelta]reg, 2]
amc1 = AdjacentMeshCells[\[CapitalDelta]reg, #, 1] & /@ mcells
amc2 = AdjacentMeshCells[\[CapitalDelta]reg, #, 2] & /@ mcells
Manipulate[
Show[
HighlightMesh[\[CapitalDelta]reg, amc2[[i]]]
, Graphics[{AbsolutePointSize[10], Red,
Point@RegionCentroid[polys[[i]]]
}]
]
, {i, 1, Length@mcells, 1}
]

You can also see the lines being highlighted by using amc1 instead of amc2 above. polys is just for centroid calculation.
MeshConnectivityGraph[Δreg, 2]. – flinty Apr 02 '22 at 12:09