Question

If the orthocentre and circumcentre of a
tringle are (-3,5),(6,2), then the centroid is​

Answer 1

Answer:If you know about Euler’s line, which I assume you dont, then the answer is quite simple.

What Euler realized when looking at those three centers of a triangle, were that they all lay on the same line, namely Euler’s line. The proof of this is not too complicated, and I suggest you watch the video by khan academy to study it:

Euler's line proof

In this proof we see that not only are the three circle centers on a line, but also that the ratio of the distances between them is always 2:1, more specifically that the distance from the orthocenter to the centroid is twice the distance from the centroid to the circumcenter. This comes from the fact that the centroid splits all the medians in the ratio 2:1, and the similar triangles found “propagate” that ratio to eulers line.

So if we define O as the origin, H as the orthocenter, C as the circumcenter and G as the centroid, we can use vectors to find out where the centroid is.

OG−→−=OC−→−+13CH−→−  

OG−→−=OC−→−+13(OH−→−−OC−→−)  

OG−→−=[62]+13([−35]−[62])  

OG−→−=[62]+13[−93]  

OG−→−=[62]+[−31]  

OG−→−=[33]  

G=(3,3)

Step-by-step explanation: