Help me with euclidean geometry

[wispy obsidian]

So I've done only problem a and haven't made any progress on b or c. Our lesson was about Thales' theorem, similar triangle.
I'm trying to figure out how to prove the two triangles IJO and BJD are similar. I dont have any other approaches in mind.

[gleaming temple]

Holy...

[wispy obsidian]

this is what i've done so far, its in vietnamese...

[gleaming temple]

Crab, you are 9th grader right?

[wispy obsidian]

yeah

howd u know

[gleaming temple]

Well, only 9th grader has to face this thing in Vietnam

[wispy obsidian]

yea lol, this really sucks

[gleaming temple]

I will try do to this

[wispy obsidian]

uh also EFBC is a cyclic quadrilateral, idk if that helps but i havent done anything with it

[gleaming temple]

It's been 3 years since I did any of this crab

[wispy obsidian]

please help me

[wispy obsidian]

oh did u js graduate from highschool?

[gleaming temple]

nah, I'm 12th grader

I'm also vietnamese has I mentioned it?

[wispy obsidian]

oh nice

u prolly in a gifted highschool right?

[gleaming temple]

can you share your diagram

I'm kinda lazy to make one

[wispy obsidian]

uh yea

https://www.geogebra.org/geometry/uudfj5ua

uh so its the one below

i drew 2 diagram, one for another problem

[gleaming temple]

so you need help with part c right?

[wispy obsidian]

yea and also b

[gleaming temple]

!xy

[tacit glacier]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[gleaming temple]

What even is I?

also

[wispy obsidian]

wat

[gleaming temple]

There is no way any triangle include BC has J as incenter

[wispy obsidian]

i think its midpoint of AH

[gleaming temple]

oh wait

It can

but what's I

[wispy obsidian]

this is the orignial problem

I is midpoint of AH

i forgor

[gleaming temple]

"trực tâm" is orthocenter

not incenter

[wispy obsidian]

oh

my bad

[gleaming temple]

I maybe too old for this lol

[wispy obsidian]

aw man

its alright

thank youu

[glossy basin]

b can be done simply with brocard

for c, N is the I-humpty pt of IBC, so its reflection is on IBC's symmedian

[wispy obsidian]

oh cool, thanks man

i have no idea what either of those are…

[green wasp]

b. Lemma: Point H is the orthocenter in triangle $ABC$ iff it lies on altitude $AD$ and $DH\cdot DA = DB\cdot DC$ (exercise to prove)
Here, we want to prove $DH\cdot DA$ = $DJ\cdot DI$ where $DH \cdot DA = DB \cdot DC$ therefore J is the orthocenter of IBC

[river locust]

v2successor

[glossy basin]

c. Basically ΔIMB～ΔBMN and ΔIMC～ΔCMN (left as exercise). With angle chasing, we know BICP is cyclic, so ∠BIP=∠BCP=∠MIC.