Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

enter image description here

share|cite|improve this question
up vote 0 down vote accepted

$OM = \frac{OA + OB + OC}3$ 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 $OA = OM - (OM - OA)$.

Hope that helps,

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.