geduiwelh
2021-01-13
Answered

(Euler line) Prove that the orthocenter M, the center O of the circumscribed circle and the barycenter S are collinear. The point S divides the segment OM in the ratio 1:2.

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Cristiano Sears

Answered 2021-01-14
Author has **96** answers

Step 1

We know that, the line on which orthocenter, circumcenter and barycenter(centroid) lie is called Euler Line of the triangle.

Now let a triangle

Here O is orthocenter, M is circumcenter and S is barycenter.

Here we can say

And also

Hence

In the above figure O is the circumcenter of

Now we have to prove O, S and M are collinear i.e., O, S and M are at the same line.

Step 2

To prove O, S and M are collinear,

we have to prove

Since

So alternate interior angle of transversal, when transversal intersect two parallel lines are congruent.

So

also we know that centroid S, splits the median into 2:1 ratio.

that is

CS=2FS

here M is the orthocenter of

and

So we have CM=2FO

Hence we can say

So we can say

Hence we can say O, S and M are collinear and S divides the line segment MO in the ratio 1:2.

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