Hello all, i had a question.
, you: )

answer: the circumcenter is the point of intersection of any two perpendicular bisectors of the triangle. meanwhile, the centroid point is the intersection of any two medians. in conclusion, if we can prove that median and perpendicular bisector are the same for the equilateral triangle, the centroid will automatically become the circumcenter. (:

step-by-step explanation:

let me know if you need any details or

Given: an equilateral triangle abc, medians ap, bq, & cr. their point of concurrency is o, which is the centroid of the triangle.
to prove that: centroid o is the circumcentre of the triangle abc.
if we prove that centroid o is the circumcentre of the triangle, then it automatically becomes the centre of the circumcircle.
proof: since ap is median, so p is mid point of bc.
ie, bp = pc.
ab = ac ( as triangle abc is equilateral)
ap=ap ( common side)
hence triangle abp is congruent to acp( by sss congruence criterion)
=> angle apb = angle angleapc ( corresponding parts of congruent triangles)
but their sum = 180°
so, each angle has to be 90°.
that shows that ap is perpendicular bisector of bc.
similarly, pyove that bq & cr are perpendicular bisectors of ac & ab respectively.
so now, the point of concurrency ‘o' of these perpendicular bisectors becomes circumcentre of the triangle. ( as circumcentre is the point of concurrency of 3 perpendicular bisectors of the sides of the triangle). and this centre is also the centre of circum circle.
this way centroid o coincides with circumcentre o
[hence proved]

That one is confusing