Show [TA] xx [AB]=[TB] xx [BC]=[TC] xx [CA] by vectors?

chivistaelmore 2022-07-18 Answered
Let T be the centroid of a triangle A B C
How can I prove that (using vectors):
T A × A B = T B × B C = T C × C A
( × is for vector-product).
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Answers (1)

Alden Holder
Answered 2022-07-19 Author has 15 answers
Noting that T A = A + B + C 3 we get [ T A ] × [ A B ] = ( A T ) × ( B A ) = ( A ( A + B + C 3 ) ) × ( B A ) = ( 1 3 ) ( 2 A B C ) × ( B A ) = ( 1 3 ) ( A × B + B × C + C × A ) Similarly, we get [ T B ] × [ B C ] = ( 1 3 ) ( A × B + B × C + C × A ) and [ T C ] × [ C A ] = ( 1 3 ) ( A × B + B × C + C × A ) which proves the claim.

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