(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.

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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Expert Answer

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 ABC and its medial triangle is DEF
image
Here O is orthocenter, M is circumcenter and S is barycenter.
Here we can say ABC is similar to DEF because D, E and F are the midpoint of lines BC, AC and AB respectively.
And also
BCEF
ACFD
ABED
Hence
ABCDEF with 2:1 ratio.
In the above figure O is the circumcenter of ABC which is also the orthocenter of DEF.
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
FOSCMS
Since CXAB and FY is the perpendicular bisector of AB. So we can say
CXFY
So alternate interior angle of transversal, when transversal intersect two parallel lines are congruent.
So
SFO=SCM
also we know that centroid S, splits the median into 2:1 ratio.
that is
CS=2FS
here M is the orthocenter of ABC and O is the orthocenter of DEF.
and ABCDEF with 2:1 ratio.
So we have CM=2FO
Hence we can say
FOSCMS by SAS similarity.
So we can say
FSO=CSM
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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