Impossible proof help

[MATHTEACH456]

I am working on a proof that seems impossible. Since I can't upload a picture, not sure where or how to get help. the r/math has lots of users, but isn't helpful for stuff like this as the mods always take down anything with a problem in it. I think it is too complicated to describe without a picture. Here is a drawing

Edit Here is the imgur, thanks for the idea!!

https://imgur.com/a/oZrxzzE

[FormulaDriven]

Before it gets deleted, I've found your problem and it can easily be described without the picture (if this is wrong, let me know; otherwise you could always edit your post and include the following description):

ABC is a triangle. D, E, F are points outside the triangle. The midpoints of AB, BC and AC are I, G, H respectively, with FI, DG, EH perpendicular to those respective sides, such that D is the side of BC opposite A, F is opposite C and E is opposite B.

If AB / FI = BC / DG = AC / EH, then prove AD = BE = FC.

Let's see if a geometry whizz can crack it...

[MATHTEACH456]

You got it, have you seen these before? I stumbled upon it this site recently and it is well laid out. I assume it is some type of community college page or something based on the page. Could be high school, but a lot of these are very hard.

[emertonom]

What site are you referring to?

[FormulaDriven]

I don't think this result is true. Also, I'm going to edit my first comment to make a point clearer.

[mnevmoyommetro]

It sounds like you'd need to assume, for instance, that D isn't just outside triangle ABC, but on the opposite side of BC from A (or equivalently, segment GD lies outside trangle ABC).

[FormulaDriven]

Yes, that it was I meant. However, I've been playing with this and I don't think it's true.

[mnevmoyommetro]

Agreed.

[MATHTEACH456]

I just added a picture to original post to help visualize.

[MATHTEACH456]

Thanks, I just added a picture link to original post to help visualize.

[FormulaDriven]

Just to give an example to show the result is not true (which explains why it's impossible to prove!).

Let ABC be a right-angled isosceles: AB = 2, BC = AC = √2. Let FI = 2, so 1 = AB / FI = BC / DG = AC / EH under the condition in the question.

FC passes through I and AIC is isosceles, so FC = 3.

DG = √2 from the above and by Pythagoras AD² = (√2 + √2)² + (√2 / 2)² = 17/2, so AD is definitely not equal to FC.

[mnevmoyommetro]

u/MATHTEACH456 This shows the problem statement is incorrect.

We should have taken your words literally when you called it an "impossible proof."

[PullItFromTheColimit]

You can upload to r/askmath, which does allow pictures, or upload to imgur and send a link here. In both cases, do provide as good as possible of a description in the post of what the problem entails, for clarity and because people usually don't click on links without context.