# A simple question about vector and geometry

My question is why vector$OM={1\over 3} (OB+OC+OD)$and $OA '$ can be expressed as the form $2OM-OA$

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$OM = \frac{OA + OB + OC}3$ because the midpoint of a polyhedron is the "average" of those points, thus you sum them and divide by their quantity (which is here $3$).
The reason why you can put $OA' = 2 OM - OA$ is because you can write $$OA = OM - (OM - OA).$$ Therefore "reflecting through the point" may be see as $$OA' = OM + (OM - OA) = 2 OM - OA.$$ Since the tetrahedron is reqular, the vector $OA - OM$ is orthogonal to the triangle $BCD$, so that the reflection of the point $OA$ to the otherside just corresponds to changing the $-$ for a $+$ in the writing of $OA = OM - (OM - OA)$.