Exploring Medial Triangles

So I've got an
arbitrary triangle here.

We'll call it triangle ABC.

And what I want to do
is look at the midpoints

of each of the sides of ABC.

So this is the midpoint of
one of the sides, of side BC.

Let's call that point D. Let's
call this midpoint E. And let's

call this midpoint
right over here F.

And since it's the
midpoint, we know

that the distance between BD
is equal to the distance from D

to C. So this distance is
equal to this distance.

We know that AE is equal
to EC, so this distance

is equal to that distance.

And we know that
AF is equal to FB,

so this distance is
equal to this distance.

Instead of drawing medians
going from these midpoints

to the vertices,
what I want to do is

I want to connect these
midpoints and see what happens.

So if I connect them, I
clearly have three points.

So if you connect three
non-linear points like this,

you will get another triangle.

And this triangle that's formed
from the midpoints of the sides

of this larger triangle-- we
call this a medial triangle.

And that's all nice
and cute by itself.

But what we're going
to see in this video

is that the medial
triangle actually

has some very neat properties.

What we're actually
going to show

is that it divides any triangle
into four smaller triangles

that are congruent
to each other,

that all four of these triangles
are identical to each other.

And they're all similar
to the larger triangle.

And you could think
of them each as having

1/4 of the area of
the larger triangle.

So let's go about proving it.

So first, let's focus
on this triangle down

here, triangle CDE.

And it looks similar
to the larger triangle,

to triangle CBA.

But let's prove it to ourselves.

So one thing we can say is,
well, look, both of them

share this angle
right over here.

Both the larger triangle,
triangle CBA, has this angle.

And the smaller triangle,
CDE, has this angle.

So they definitely
share that angle.

And then let's think about
the ratios of the sides.

We know that the ratio of CD
to CB is equal to 1 over 2.

This is 1/2 of this entire
side, is equal to 1 over 2.

And that's the same thing
as the ratio of CE to CA.

CE is exactly 1/2 of CA,
because E is the midpoint.

It's equal to CE over CA.

So we have an angle,
corresponding angles that

are congruent, and
then the ratios

of two corresponding sides
on either side of that angle

are the same.

CD over CB is 1/2, CE over CA
is 1/2, and the angle in between

is congruent.

So by SAS similarity, we
know that triangle CDE

is similar to triangle CBA.

And just from that, you can
get some interesting results.

Because then we
know that the ratio

of this side of the smaller
triangle to the longer triangle

is also going to be 1/2.

Because the other two
sides have a ratio of 1/2,

and we're dealing with
similar triangles.

So this is going
to be 1/2 of that.

And we know 1/2 of AB is just
going to be the length of FA.

So we know that this
length right over here

is going to be the
same as FA or FB.

And we get that straight
from similar triangles.

Because these are similar,
we know that DE over BA

has got to be equal
to these ratios,

the other corresponding
sides, which is equal to 1/2.

And so that's how we got
that right over there.

Now let's think about
this triangle up here.

We could call it BDF.

So first of all, if
we compare triangle

BDF to the larger
triangle, they both share

this angle right
over here, angle ABC.

They both have that
angle in common.

And we're going to have
the exact same argument.

You can just look
at this diagram.

And you know that the ratio
of BA-- let me do it this way.

The ratio of BF to
BA is equal to 1/2,

which is also the
ratio of BD to BC.

The ratio of this
to that is the same

as the ratio of this
to that, which is 1/2.

Because BD is 1/2 of
this whole length.

BF is 1/2 of that whole length.

And so you have
corresponding sides

have the same ratio
on the two triangles,

and they share an
angle in between.

So once again, by
SAS similarity,

we know that triangle--
I'll write it

this way-- DBF is
similar to triangle CBA.

And once again, we use this
exact same kind of argument

that we did with this triangle.

Well, if it's similar, the ratio
of all the corresponding sides

have to be the same.

And that ratio is 1/2.

So the ratio of this
side to this side,

the ratio of FD to
AC, has to be 1/2.

Or FD has to be 1/2 of AC.

And 1/2 of AC is just
the length of AE.

So that is just going to be
that length right over there.

I think you see
where this is going.

And also, because it's similar,
all of the corresponding angles

have to be the same.

And we know that
the larger triangle

has a yellow angle
right over there.

So we'd have that yellow
angle right over here.

And this triangle
right over here

was also similar to
the larger triangle.

So it will have that same
angle measure up here.

We already showed that
in this first part.

So now let's go to
this third triangle.

I think you see the pattern.

I'm sure you might be able
to just pause this video

and prove it for yourself.

But we see that the
ratio of AF over AB

is going to be the
same as the ratio of AE

over AC, which is equal to 1/2.

So we have two corresponding
sides where the ratio is 1/2,

from the smaller
to larger triangle.

And they share a common angle.

They share this angle in
between the two sides.

So by SAS similarity--
this is getting repetitive

now-- we know that triangle
EFA is similar to triangle CBA.

And so the ratio of all
of the corresponding sides

need to be 1/2.

So the ratio of FE to
BC needs to be 1/2,

or FE needs to be 1/2 of that,
which is just the length of BD.

So this is just going to be
that length right over there.

And you can also
say that since we've

shown that this triangle, this
triangle, and this triangle--

we haven't talked
about this middle one

yet-- they're all similar
to the larger triangle.

So they're also all going
to be similar to each other.

So they're all going to have
the same corresponding angles.

So if the larger triangle
had this yellow angle here,

then all of the
triangles are going

to have this yellow
angle right over there.

And if the larger triangle
had this blue angle

right over here, then in
the corresponding vertex,

all of the triangles are
going to have that blue angle.

All of the ones that
we've shown are similar.

We haven't thought about this
middle triangle just yet.

And of course, if this
is similar to the whole,

it'll also have this
angle at this vertex

right over here, because this
corresponds to that vertex,

based on the similarity.

So that's interesting.

Now let's compare the
triangles to each other.

We've now shown that
all of these triangles

have the exact same three sides.

Has this blue side-- or
actually, this one-mark side,

this two-mark side, and
this three-mark side.

One mark, two mark, three mark.

One mark, two mark, three mark.

And that even applies
to this middle triangle

right over here.

So by side-side-side
congruency, we now know--

and we want to be careful to get
our corresponding sides right--

we now know that triangle CDE
is congruent to triangle DBF.

I want to get the
corresponding sides.

I'm looking at the colors.

I went from yellow to magenta
to blue, yellow, magenta,

to blue, which is going to
be congruent to triangle

EFA, which is going to be
congruent to this triangle

in here.

But we want to make
sure that we're

getting the right
corresponding sides here.

So to make sure we
do that, we just

have to think about the angles.

So we know-- and
this is interesting--

that because the interior
angles of a triangle

add up to 180 degrees,
we know this magenta

angle plus this blue angle plus
this yellow angle equal 180.

Here, we have the blue
angle and the magenta angle,

and clearly they will
all add up to 180.

So you must have the blue angle.

The blue angle must
be right over here.

Same argument-- yellow
angle and blue angle,

we must have the magenta
angle right over here.

They add up to 180.

So this must be
the magenta angle.

And then finally,
magenta and blue-- this

must be the yellow
angle right over there.

And so when we wrote
the congruency here,

we started at CDE.

We went yellow, magenta, blue.

So over here, we're going
to go yellow, magenta, blue.

So it's going to be
congruent to triangle FED.

And so that's pretty cool.

We just showed that all
three, that this triangle,

this triangle, this
triangle, and that triangle

are congruent.

And also, we can look
at the corresponding--

and that they all have
ratios relative to-- they're

all similar to the larger
triangle, to triangle ABC.

And that the ratio between
the sides is 1 to 2.

And also, because we've looked
at corresponding angles,

we see, for example,
that this angle

is the same as that angle.

So if you viewed DC
or if you viewed BC

as a transversal,
all of a sudden

it becomes pretty clear that FD
is going to be parallel to AC,

because the corresponding
angles are congruent.

So this is going to be parallel
to that right over there.

And then you could use
that same exact argument

to say, well, then this
side, because once again,

corresponding angles
here and here-- you

could say that
this is going to be

parallel to that
right over there.

And then finally, you make
the same argument over here.

I want to make sure I get the
right corresponding angles.

You have this line
and this line.

And this angle
corresponds to that angle.

They're the same.

So this DE must
be parallel to BA.

So that's another neat property
of this medial triangle,

[? I thought. ?]
All of these things

just jump out when you just try
to do something fairly simple

with a triangle.