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