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

chivistaelmore

chivistaelmore

Answered question

2022-07-18

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).

Answer & Explanation

Alden Holder

Alden Holder

Beginner2022-07-19Added 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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?