Math Problem Geometry

[wary reef]

The vertex M of the equilateral triangle CMD belongs to the circle with the center O, and the vertex C D to the chord AB of this circle. It is known that AC = CD = DB. Find the angle of the AOB.

Here’s my picture it’s kinda bad but hope it helps I translated the question from Russian I hope it still makes sense.

[odd garden]

gonna analyze it a little

so what can you make of triangle MCD being equilateral?

[wary reef]

Make like I know I can pull a line from M to the middle of CD and it will be 90 degrees

I can pull AM and AB and the triangles will be 2 same sides

Like they will be equal

[odd garden]

problem is you don't know the angular properties of AB

so if MCD is equilateral how big are each angle?

[wary reef]

Yeah like they are the same

[odd garden]

all 3 are the same

[wary reef]

Like angles ACM and ADB

They are also the same

[wary reef]

All 60

[odd garden]

yep

[wary reef]

But that doesn’t help us

[grave fern]

Adding these will help

[odd garden]

calculate all the angles if you can

[wary reef]

Like yeah Ik that these are 120

[odd garden]

what does angle AMC equal to

[grave fern]

Ok so what does that say abt the central angle

[wary reef]

I know this

Already tried that

[odd garden]

what is angle AMB then

[wary reef]

120

[odd garden]

120 right?

[wary reef]

Yep

[grave fern]

Yes this is the right way

[wary reef]

But that doesn’t mean AOB=AMB

That’s the problem

[odd garden]

now you just mentioned a great idea

[wary reef]

No bro

Doesn’t mean O is on that line

[odd garden]

you can connect M to O directly

[grave fern]

It’s aob = 2*amb

[wary reef]

No wth

[wary reef]

No why

[odd garden]

O is actually on that line

[wary reef]

No proof

[wary reef]

Why

Tell me

[odd garden]

you can prove it

so OA=OB

[wary reef]

Could be like this

[odd garden]

ye (the line is a little off)

[wary reef]

That’s the point

It can be off

There is no proof MO is deciding CD and is 90

*devieding

[odd garden]

take the midpoint of AB

[wary reef]

That’s the same

But doesn’t mean MO is deviding it

[wary reef]

The triangles are not proven equal

[odd garden]

I think you haven't learnt about inscribed angles yet

[grave fern]

It’s the same arc

[wary reef]

I know that that’s it the 180-the other angle

[wary reef]

No why

Triangle AOB is not AMB

what now

[odd garden]

^ @grave fern

[grave fern]

Arc AB

[grave fern]

Ohh

[wary reef]

What ant it

Abt it

Yeah AB is equal for both

Doesn’t mean anything

Im 8th grade wth

[odd garden]

I actually didnt learn about this until 9th grade along with cyclic quads

so

[wary reef]

Yup

[grave fern]

Ts is how I did the q

[grave fern]

But if he didn’t learn abt these there must be another way then

[wary reef]

Bro

[odd garden]

there is

using special properties of 60° angles

it just so happens triangle AOB and AMB are congruent in this case

[grave fern]

Yea

Ok sure

[odd garden]

so you have to take the midpoint of AB

[wary reef]

I’ll send a vid @grave fern

[odd garden]

call it H

[wary reef]

Did

[grave fern]

U misunderstood I wasn’t talking abt the triangles

[wary reef]

Huh

[odd garden]

can you prove triangle MHA and MHB are congruent?

[wary reef]

Wha does congruent mean

[odd garden]

the triangles are the same

[wary reef]

Like two same sides

[grave fern]

Nvm u don’t seem to have learned inscribed angles

[wary reef]

Yeah

I can

[odd garden]

so what angles are MHA and MHB

@grave fern I remember we have to do long proofs in these cases lmao

[odd garden]

in grade 6 to 8

[odd garden]

^

[grave fern]

lol I just saw this way was like so short but if he hasn’t learned it he won’t understand it

[wary reef]

Yeah idk how to do

[odd garden]

connect M to H

actually did you learn about perpendicular bisectors?

[grave fern]

What did u assign h to ? mo?

[wary reef]

Did

[grave fern]

Oh mid point kk

[odd garden]

midpoint of AB

[wary reef]

Doesn’t help

[odd garden]

oh then you don't need H then

so if OA = OB, does O lie on AB's perpendicular bisector?

[wary reef]

Ur Chinese

How do u not know this

[odd garden]

no im not

[wary reef]

Huh

[odd garden]

I'm vietnamese

ignore my about me

[wary reef]

It isn’t

No proof

[odd garden]

it's just full of vietnamese rhetoric nonsense

[wary reef]

Ok

[odd garden]

so you got MHA=MHB=90

now connect O to H

can you prove triangle OHA and OHB are the same as well?

[grave fern]

They are both R AO=OB=R (R is the circle’s radius)

[odd garden]

ye and my idea is to prove O, H and M lie on a line

[grave fern]

Yea I realized

[odd garden]

and use it to prove OAM and OBM are equilateral

painful without better geometry tools

[grave fern]

If oh cut ab in half so does hm that would mean h is on om

[odd garden]

yea because you create more right angles

that add to 180

[grave fern]

Yea

[grave fern]

Mhm so 60+60 =120

[odd garden]

dw I have seen much worse in 9th grade

my 9th grade geometry is a monster to say the least

[grave fern]

U sound traumatized loll

[odd garden]

then I chose chem class in grade 10 which clears everything away

[grave fern]

I majored in physics n cs

[odd garden]

almost got to vietnamese IChO team

[grave fern]

Didn’t have a traumatic experience with 9th grade math lol

[odd garden]

failed miserably

now I get back to math

[grave fern]

That’s cool

[grave fern]

Where did the kid go?

[odd garden]

idk

[grave fern]

Loll

[odd garden]

geometry is traumatic sometimes

[grave fern]

Ngl it’s my fav

[odd garden]

like there are so many shit i had to remember to even get a chance to get into this school

Miquel's theorem, Reim's theorem

[grave fern]

Well the more u solve it just sticks to ur mind yk

[odd garden]

I didnt have a proper geometry teacher until then

he was the sole reason I fell in love with geometry

[odd garden]

before him you have to throw everything into a wall and see what sticks

[grave fern]

Thats sweet lowk

I’m Finally graduating 😎

[odd garden]

which is why it's common to see kids practicing 50 60 geometry problems just to memorize and get into a good school in grade 10

[grave fern]

I can learn whatever I want

[odd garden]

same here

[grave fern]

Ooh ic

[odd garden]

vietnamese education system is fucked

[grave fern]

07s best gen fr

[odd garden]

idk about the case in the us

[grave fern]

Oh

[grave fern]

Same

I’m not American lol

[odd garden]

this year they decide to remove cyclic quads in grade 9 and complex numbers in grade 12

ridiculous

[grand oasis]

ur an entrant?

[grave fern]

Nah complex numbers is my sec fav

They did yall dirty

[odd garden]

cyclic quads are so important

[grave fern]

No?

[grand oasis]

did op solve the problem

[grave fern]

He disappeared

[odd garden]

got traumatized

[grand oasis]

🙂

[grave fern]

[grand oasis]

kidnapped by geometry

[odd garden]

geometry monster brainrot

popular meme here

[west flax]

am love geom

[viral canopy]

it would be so awesome if i could prove there was a radius that passes through both of them (entirely)

[grand oasis]

would be nice if it was used in higher mathematics

[odd garden]

yea

but geometry apparently is more sinister than satan

[grave fern]

Yo don’t come for geometry like dat

[grave fern]

Immaculate

[west flax]

indeed

[viral canopy]

i didn't even learn complex numbers in any grade until i studied them on my own

de moivre's theorem was fun

[odd garden]

me still reading euclidean geometry papers in 2025:

[viral canopy]

didn't know people still wrote about it

euclid published ts in 200 bc and people still be using it n shiet

[odd garden]

ye 2014 some guy wrote about ERIQ theorem on AoPS

and Vietnamese mfs start spamming it in olympiads 😭

I remember there's a problem so horrid that they have to prove 10 lemmas along the way

[wary reef]

How can u prove thatMH =HO

[odd garden]

you can't for now

prove triangle OHA and OHB are the same first

[wary reef]

U can

[odd garden]

so answer me

does OA=OB?

[wary reef]

Yes

[odd garden]

does HA=HB?

[wary reef]

Yes

[odd garden]

and those share the side OH right?

so are those two the same?

[wary reef]

Yes

The triangles are

[odd garden]

good

[wary reef]

But how are oh=hi

[odd garden]

so what angle is OHA and OHB

[wary reef]

Hi

Same

90

[viral canopy]

y all ever notice mdc is just 1/3rd of aob

[viral canopy]

i 'll go work on this privately rq

[wary reef]

Ok

[odd garden]

so what is the total angle of OHA and MHA

[wary reef]

180

[odd garden]

so does H lie on OM?

[wary reef]

Yeah I already proved that

[odd garden]

good

[wary reef]

I only need MH=OH

[odd garden]

now what angle is AMO?

[wary reef]

90

[odd garden]

try again

[wary reef]

90

30+60

[odd garden]

you need to add each component angle

AMC+CMH

AMC is 30

[wary reef]

Yeah90

[odd garden]

what is CMH?

[wary reef]

30

Oh

60

[odd garden]

so AMO is 60 right?

[wary reef]

Yep

[odd garden]

so look at triangle AMO

you got OA=OM

You got one angle =60

what does it tell you

[wary reef]

No

Yearning do

Yes

I do

[grave fern]

Well a shortcut would be using the properties of a kite shape and the isosceles triangle

[odd garden]

what does it tell you about triangle AMO then?

[grave fern]

Like so

[odd garden]

nice little kite ngl

[grave fern]

But ur already doing another thing so

[grave fern]

I draw the shape to remember its properties hehe

[odd garden]

nice use of 30-60-90s

I like that

[grave fern]

Mhm

[wary reef]

I proved

[grave fern]

Not using the relationship between an inscribed angle and a central angle is criminal I can’t get over it

[odd garden]

nice job

[odd garden]

it's like having to drop the most useful tool

[grave fern]

Yea

[wary reef]

I Habenseite math problem ready

Have

Next

Prove that if the triangle is acute, then the center of its circumscribed circle lies inside the tragol, and if the triangle is obtuse, then it is outside the triangle.

@grave fern @odd garden

[odd garden]

why am I scared of this problem lmao

[grave fern]

Which triangle r u referring to ?

[odd garden]

any acute or obtuse triangle

it's a general fact

[grave fern]

Oh ok

I thought he wanted us to prove it from a triangle from that q

[odd garden]

proving inside/outside a triangle maybe a problem

I remember only given this as a fact

I never had to prove sth like this

[grave fern]

If its right angled the center is on the triangles leg

[odd garden]

ye

[grave fern]

Same

[grave fern]

What if its like using the relation that says if an inscribed angle =90 its arc is the diameter ?

[odd garden]

i think I got an idea

[wary reef]

Show me bbg

[odd garden]

which uses this

but it's not really rigurous

[grave fern]

But he doesn’t know inscribed angles loll

[odd garden]

I think I might have an idea

[grave fern]

There is also one that describes the relation between the arc and its inscribed angle like the bigger the angle the taller the arc the closer to the center idk whats it called in English

Its so hard using eng terminology ik these in arabic loll

[wary reef]

Ur Arabic Christian?

That’s very tuff

[grave fern]

Yes arab Christian

[wary reef]

Catholic?

[grave fern]

Yes

[wary reef]

Omg so tuff

Im Russian catholic

[odd garden]

oh no

proofs I see online all use inscribed angles

[wary reef]

Lol

What is an inscribed angle

Yeah I looked it up pure shit

[grave fern]

Cool

[grave fern]

Um

[odd garden]

even the math forums have really complicated solutions

[grave fern]

Tbh i dont think this is a homework question

[odd garden]

can't even understand them

[ivory latch]

this problem isn’t bad imo you can call the radius of the circle r, call the side length of the equilateral triangle s, and do some simple geometry to get side lengths of a right triangle that, using trig, can tell you the angle you’re looking for

[odd garden]

he hasnt learnt trig yet

[ivory latch]

oh

[odd garden]

that's the problem

[ivory latch]

he should because trig is fun

[odd garden]

trig is my biggest enemy

[grave fern]

Yea proving is not the problem what he knows is

[odd garden]

well, alongside inequalities

[grave fern]

My teacher when first taught ts I remember she proved it using trig

[grave fern]

Well i hate anything that involves memorizing history…

Like listening and learning about it tho

As for math currently there isn’t much i hate

[odd garden]

nothing can beat inequalities for me

it's the worst thing in existence for a person who is slow at transforming shit in algebra

I skipped all inequalities in my grade 10 entry exam

[grave fern]

Oh wow

[odd garden]

I bet 97% of us skipped

inequalities is a monster of its own

it is pure trauma and agony

[torpid pebble]

@wary reef did you guys solv it ?

[odd garden]

ignore this

[grave fern]

Dows it have time limit? Like 1min

Cuz thats so longgg

[odd garden]

this is from my school's grade 10 entry test this year

[ivory latch]

I figured out a neat way to solve it with no trig

[odd garden]

6 exercises, 150 minutes

[odd garden]

show it

[ivory latch]

I can’t just post the answer can I

I mean I have the work too

[odd garden]

you can

[ivory latch]

although it might be hard to decipher what I did

[odd garden]

we don't know how to solve this without trig as well as at school we are only taught this as an "obvious" fact

[grave fern]

Yea 25min is reasonable

[odd garden]

uhhh not to this problem

[grave fern]

In general

[odd garden]

this is the problem we're struggling at

@ivory latch

that one we've already solved

[ivory latch]

didn’t even see that problem

[odd garden]

wanna see solution?

[grave fern]

To the inequality?

[odd garden]

yea

[grave fern]

Wait let my try first

Ill try it in ten min when i get home

[odd garden]

alr

honestly I should buy a book about inequalities

[ivory latch]

thales’s theorem might be useful

if the triangle is a right triangle, the center of the circle lies on its hypotenuse

then you could show if you increase the largest angle, the center is no longer inside the triangle, and if you decrease it the circle is inside the triangle

I can’t do proofs tho so you guys can figure out the formalities

[grave fern]

Yea i said this too hehehe

[grave fern]

Btw why is it so long thats just annoying atp loll

[odd garden]

don't ask me, ask the monster who made this

[grave fern]

Lolll

Idont even wanna bother is there a short cut or do i have to expand it all

[odd garden]

I don't want to get any more ptsd from this

[odd garden]

Do not expand

for the love of god

[grave fern]

Yea i wont

Thats tooo long

I never got qs like these with three variables

[odd garden]

this appears all the time

[odd garden]

tbf I could've solved one of them if I remember Abel's sum

I even used it to solve a problem in my grade 10 first term exam

[grave fern]

Yea

[grave fern]

@wary reef if ur done please .close ts

[wary reef]

.close