let’s mark up the feet of the altitudes
of any triangle the midpoints of these
sides and the midpoints of the segments
from the vertices to the earth’s center
do you know why this composition is so
impressive
it seems that this one and many other
beautiful theorems were well known in
ancient greece a long time ago and all
we are left to do is reap the ancient
fruits but no the nine-point circle was
discovered by euler and lots of other
surprising facts were found in the 20th
century for example morley’s trisector
theorem the picture shows angle
trisections what do you think makes this
highlighted triangle stand out
it is always equilateral frank morley
came across this wonderful property when
he was studying cubic curves one of the
most concise proofs of this theorem was
created by john conway a couple of
decades ago of course he has plenty of
other results but today i also want to
mention the conway circle sketch an
arbitrary triangle and extend its sides
pay attention to the marks of equal
segments eventually we will have six
points on one circle its center always
coincides with the center of the
inscribed circle here is another theorem
that slipped away from euclid and even
ptolemy let’s look at the circumscribed
circle of an equilateral triangle
connect a point from one of the arcs
with the vertices of the triangle notice
anything particularly interesting if you
don’t then look at the lengths of the
segments and you will probably figure it
out surprisingly pompeo’s theorem was
discovered only in 1936
let’s prove it follow the bottom
triangle rotating it 60 degrees around
point a vertex c will move into b
because the angle of an equilateral
triangle is exactly 60 degrees and point
b will inevitably turn into e on bp in
the comments try to explain why it is
happening and the theorem is proved when
we rotate segment ip it turns into equal
segment ae angle b i
is equal to the rotation angle it means
that triangle ape is equilateral so the
sum ap plus pc is equal to the sum pe
plus be and this is actually the length
of bp
by the way do you remember the theorem
named after napoleon
we will certainly prove it in one of my
videos later as far as i’m concerned
similar fact about quadrangles was
published only in the 20th century and
was called thibault’s theorem let’s draw
outside squares on the sides of the
parallelogram then the centers of these
squares form what
right another square do you want one
more marvelous proof based on rotation
let’s look at the triangle lbm and
rotate it 90 degrees around point m what
do you think we will get in the end oh
wow it moved into triangle and cm but
why think about it regarding bm and mc
sides everything is clear they are equal
since they are diagonals of the square
the same happens to be l and c n
all you are left to do is show that
angles lbm and mcn are equal try it and
if you want succeed then look at the
description of the video eventually
segments lm and nm are equal because
they overlap each other after rotation
the same approach works for other
triangles and it means that in the
quadrangle that is particularly
interesting for us all sides are equal
and all angles are right so it is a
square
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please notice that if we draw squares on
the sides of an arbitrary quadrangle
then diagonals still be equal and will
cross each other at right angle this
theorem was previously proved by van
able and in the 20th century more
general fact was found
finally let’s draw the medians of a
triangle
you probably know why they divide it
into six parts with equal areas but in
2000 a new and fantastic problem
appeared i will leave it to the
animation think critically do math take
care
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you