What I wanna do on this video, for some triangle we're gonna focused
on this larger triangle over heretriangle ABC
What I wanna do is prove that circumcenter,
remember the circumcenter is the intersection
of it's perpendicular bisectors
The circumcenter for this triangle, the centroid of this triangle,
the centroid is the intersection of it's medians,
and the orthocenter, that's the intersection of it's altitudes
all sit on the same line, or that OI right over here really a line segment
Or that OG and GI are really just 2 segments
that make up these 2 line segments which is part of the euler line
And to do that, I've set up a medial triangle right over here,
triangle FED, or actually I should say, triangle DEF,
whch is the medial triangle for ABC
And there's already a bunch of things
that we know about medial triangle
and we've proven this in previous videos
One thing we know, is the medial triangle DEF,
DEF is going to be similar to the larger triangle
the triangle that is a medial triangle of,
it is of, and ratio from the larger triangle to the smaller triangle
it's a 2 to 1 ratio
And this is really important to proof
When two triangles are similar with the given ratio,
that means if you take the distance between any 2 corresponding parts
of the two similar triangles that ratio will be 2 to 1
Now the other relationship that we've already shown,
the other relationship between the medial triangle,
and the triangle that is the medial triangle
of is that we've shown the orthocenter
of the medial center of the larger triangle
So one way to think about it is Point O,
we already mentioned is the circumcenter of the larger triangle
It is also the, it is also the orthocenter of the smaller triangle,
we actually wrote it of here
so Point O noticed it is on this perpendicular bisector,
you know, I should do a bunch of other ones of this dark grey color
but I didn't wanna make this diagram too messy
But this is the circumcenter of the larger triangle
Now in order to prove that OG and I all sit on the same line
or the same segment in this case
What I'm going to do, I wanna prove,
I'm going to prove that triangle FOG,
is similar to, is similar to triangle CIG,
is similar to triangle CIG
Because if I can prove that, then their corresponding angles
are going to be equivalent,
you could say this is angle is going to be equal
to those angle over here
And so OI would have to be a transversal,
cause we're goiong to see these two lines here are parallel
or if these two triangles are similar,
just remember that someone's look at our triangle here
and this triangles over there
If they truly are similar then this angle
is going to be equal to that angle
which would mean that,
so these are really would be vertical angles
and so this really would be real line
So let's go the actual proof
So maybe, I don't need those to highlight it over here
So one thing and I've hinted about this already,
we know that this line over here, we can call this like XC
we know this is perpendicular like AB
it is an altitude, and we also know that FY right over here
is perpendicular to AB, it is a perpendicular bi-sector
So they both form the same angle with a transversal
you can view AB as a transversal
so they must be parallel, so we know that FY,
FY is parallel to XC, to XC
segment FY is parallel to XC,
segment FY is parallel to segment XC
And we can write it; this guy is parallel to that guy there
And that's useful because we know that alternate interior angles
of a transversal, when a transversal intersects two parallel
lines are congruent
So we know, we know that this angle, so we know that FC is a line
it is a median of this larger triangle, triangle ABC
So you have a line intersecting two parallel lines,
alternate interior angels are congruent
So that angle is gonna be congruent to that angle
So you could say angle OFG is congruent to angle,
so it's OFG, it's congruent to angel ICG,
to ICG, now the other, the other thing we know,
this is a property, this is a property of medians is that
a median splits up or should I say the centroid,
splits the median in to two segments that have a ratio of two to one
or another way to think about this is a centroid
is two thirds along the median
So we know we've proven this on a previous video
We know that CG, CG is a equal to 2 times GF, 2 times GF
and I think you see where you are going,
we have an angle, I've shown you that the ratio of this side
to this side is 2 to 1 and that's just the property
of centroids and medians
And now if we can show you the ratio of his side CI is FO is 2 to 1
that we have two corresponding sides where the ratio is 2 to 1,
and we have the angle and between this congruent,
and we have the SAS singularity to show that these two triangles
are actually similar, so let's actually think about that
CI is the distance between, CI is the between the larger triangle's
point C orthocenter of the larger triangle
Well what is FO?
Well F is a corresponding point to point C on the medial triangle
and we make sure that we specify the similarity with the right, F,
F corresponds to point C
So FO is the distance between F on the smaller medial triangle
and the smaller medial triangle's orthocenter
So this is the distance between C
and the orthocenter of the larger triangle
This is the distance between the corresponding
side of the medial triangle
and the smaller medial triangle and it's orthocenter
So this is the same corresponding distance
on the larger triangle and the medial triangle,
and we already know that they're similar with the ratio of 2 to 1
And so the corresponding distances between any 2 points
on the two same triangle are gonna have the same ratio
So because of that similarity, because of the similarity
we know that CI, CI is gonna be equal to 2 times FO
I wanna emphasize this; C is the corresponding point to F,
when we look at both of these similar triangles,
I is the orthocenter of the larger triangle,
O is the orthocenter of the smaller triangle
You're taking a corresponding point
to the orthocenter of the larger triangle,
corresponding point of the smaller triangle
The triangles are similar to the ratios of 2 to 1
so the ratio of this length to this length is going to be 2 to 1
So we've shown, we've shown the ratio
of this side to this side is 2 to 1
We've shown the ratio if this side to side is also to 2 to 1
We've shown the angle in between are,
the angle between them is congruent
So we have proven by SAS similarity,
so it goes down a little bit, so by, by SAS similarity,
not congruency, similarity we've proven that triangle FOG
is similar to CIG
And so we know corresponding triangles are congruent,
we know that angle CIG correspond to angle FOG
so those are going to be congruent, and we also know that angle CGI,
angle CGI, let me do this a new color,
angle CGI corresponds to angel OGF
so they're also going to be congruent
So you can look at the different ways of these angle
and this angle are the same you can now view OI
as a true line as a transversal of these two parallel lines
So that let's you know that's a one line
or you can look these two over here,
so look these two angles are equivalent
so these must be vertical angles,
so this must actually be, this actually must be the same line
The angle that this is approaching, this,
this median is the same angle that is leaving
So these are all, these are definitely on the same line
So it's a very simple proof, once again,
from a very profound idea, the orthocenter,
the centroid and the median of any triangle
all sit on this magical euler's line