complex number question help needed (1 Viewer)

[let.me.die]

hi. im struggling with the geometrical interpretation of complex numbers.. i was wondering if someone could help me with these questions:

1. the centroid of a triangle is the point of intersection of the medians of the triangle. show that (z1 + z2+ z3) / 3 represents the centroid of the triangle whose vertices are z1, z2 and z3.

2. the circumcentre of a triangle is the point of intersection of the perpendicular bisectors of the three sides. find the complex number z that represents the circumcentre of a triangle whose vertices are z1, z2 and z3.

 

[who_loves_maths]

hi let.me.die,

sorry about answering this question so late... i hope it's not too tardy for you... just never saw this thread until late this arvo.

Question 1:

Question 2:

sorry about skipping the middle algebra bash part for Question 2. it was just too boring to write out properly...
also, there is probably a more simplified version for 'z' in Question 2 than that i have left it as. but i just cbb simplifying
though i'm sure it's fine that way it is, unless the question actually specifies the form the answer should be in. {but i'm sure you can simplify it yourself anyways}


anyhow, hope that helps

 

[who_loves_maths]

in fact, now that i think about it... there are most certainly better and more expedient methods out there for the solution of Question 2.
maybe i should try some circle geometric properties...

another thing worth noting here is that the formula for the circumcentre of the triangle is heavily related to the orthocentre of a triangle.
eg. join the three midpoints of each side of a triangle to form a smaller inner triangle within. now, the circumcentre of the larger triangle is equivalent to being the orthocentre of the inner triangle (whose vertices' coordinates, being midpoints, are not hard to find at all) ...

P.S. this btw, is a very simple, non-algebraic or even geometric, way of showing that, for any triangle, the three heights dropped from each vertices are concurrent about a single point - the orthocentre.