# Open Proof Contest 3

Hello Every$$\displaystyle \int_0^1{dx}$$.

This is the third part of the contest started by me.Here is the $$3^{rd}$$ part.The rules are the same as the previous ones. You need need to mail the PDFs to opencontestproofs@gmail.com. To make a PDF you can go to Latex PDF. Also you can send a snip of the solution after typing it in $$\LaTeX$$.

Here are the questions

1.Look at the figure below :

It is given that $$AB || GF$$ and $$EFGH$$ is parallelogram. Prove that $$FE || DC$$ .

$$2$$. Look at the figure below :

$$ABCD$$ is a parallelogram and $$AXY$$ is a straight line through $$A$$ meeting $$BC$$ at $$X$$ and $$DC$$ at $$Y$$. Prove that $$BX\times DY$$ is a constant.

$$3$$.Look at the figure below:

Given four points $$A,B,C,D$$ in a straight line, show that a point $$O$$ exists in the same straight line such that $$OA:OB = OC:OD$$.Also find that point.

$$4$$.Look at the figure below:

$$PQ$$ is a chord of a circle. Through the mid point $$M$$ of $$PQ$$ chords $$AB$$ and $$CD$$ are drawn. $$AD$$ and $$BC$$ meet $$PQ$$ at $$K$$ and $$L$$. Then prove that $$M$$ is the midpoint of $$KL$$.

## $\huge \textbf{Cancelled}$

Theme : Geometry Triangle

Note by Rajdeep Dhingra
3 years, 6 months ago

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I will participate $$\ddot\smile$$ $$\ddot\smile$$ $$\ddot\smile$$ $$\ddot\smile$$ $$\ddot\smile$$ $$\ddot\smile$$

- 3 years, 6 months ago

Comment deleted Apr 12, 2015

Yeah ,it's correct . Looks like you'll win this contest for sure :P

- 3 years, 6 months ago

Well , Euclid will become angry and will come for revenge on seeing my uploaded diagram above. :P :D xD

- 3 years, 6 months ago

No he won't . There's only one $$\color{red}\text{Euclidean}$$ error in it . Just add two arrow heads on the ends of the line and you're done :)

- 3 years, 6 months ago

Even I thought there was something wrong with the problem! Your proof is flawless. Just to clarify, ahead of $$E$$ being the midpoint of $$AC$$, you can say that $$E$$ was chosen to lie on $$XC$$. Hence, $$A$$ and $$B$$ are coincident and the triangle cannot be formed.

- 3 years, 6 months ago

Q4 has been updated.

- 3 years, 6 months ago

I have mad some minor mistakes while giving solutions.Please make sure you read the corrections that I have sent.

- 3 years, 6 months ago

Yup thanks! It seems pretty interesting :)

- 3 years, 6 months ago

Q4 has been updated.

- 3 years, 6 months ago

Thanks a lot!

- 3 years, 6 months ago

Welcome.

- 3 years, 6 months ago

I think I have seen this somewhere... Did you create these problems?

- 3 years, 6 months ago

This one not. It is a very famous theorem.

- 3 years, 6 months ago

I don't know its name , but I had proved this in 8th Std. But I don't remember the proof now , So, i have to rethink now :)

- 3 years, 6 months ago

- 3 years, 6 months ago

I think , it must be deleted after 24 hours , or the people who are solving or have taken note of the previous problem will start questioning 'why it is removed , where it has gone?' .Hence I prefer deleting it after 24 hours.

- 3 years, 6 months ago

Yes you are right. I am thinking of removing it. I will add another one in place of it.

- 3 years, 6 months ago

Actually this suspicion came by @Omkar Kulkarni 's median contradiction.

- 3 years, 6 months ago

Is Q4 fine ?

- 3 years, 6 months ago

Yeah its good , I got it!

- 3 years, 6 months ago

Are you going to participate ? Deadlines near

- 3 years, 6 months ago

I give up on the third one, I'll try to send the others before 10:29! I completely forgot about coding the solutions. :P

- 3 years, 6 months ago

Yeah , There was some problem in my latex , so I cannot send the solution for 3rd problem.No option :/ I will send solutions to only problem number 1,2,4.

- 3 years, 6 months ago

@Rajdeep Dhingra , When Will you give out the results? -_-

- 3 years, 5 months ago

This contest is cancelled and we all are trolled ... xD Cheers! :( :(

- 3 years, 5 months ago

Could you post a solution for the third one? It is troubling me SOO much.

- 3 years, 5 months ago

Why the hell is the Contest Cancelled? o.O

- 3 years, 5 months ago

Sorry for that. But since Schools are going on and I am very much busy, I can't Compile the result. I will release OPC 4 on 18th May. Do participate in that one. Sorry.

- 3 years, 5 months ago

Okay. No problem. :)

- 3 years, 5 months ago

Rajdeep would you suggest some books for RMO

- 3 years, 5 months ago

@Rajdeep Dhingra I had error with converting Latex to pdf.So i managed to convert Latex to photo. Is it ok if I send you photos?

- 3 years, 6 months ago

Yes. You can send the photos.

- 3 years, 6 months ago

@Rajdeep Dhingra , Can I Give my solution in MS Word? I am not really familiar with LaTeX Yet. Also, Can I give my solution in Parts? I have a lot of Homework To do.

- 3 years, 6 months ago

Yes, Of course.

- 3 years, 6 months ago

Okay :)

- 3 years, 6 months ago

I will participate

- 3 years, 6 months ago

- 3 years, 6 months ago

All the best !

- 3 years, 6 months ago

Could you provide a link to the previous ones, so as to give everyone easy access to the rules? Thanks. I'll participate too :)

- 3 years, 6 months ago

- 3 years, 6 months ago

- 3 years, 6 months ago

Where is G in the figure to the first question ?

- 3 years, 6 months ago

G is the point on the parallelogram. GH is parallel to EF

- 3 years, 6 months ago

I understood that , but won't you make an edit in the picture ?

- 3 years, 6 months ago

Right now I have to go somewhere. So I will make it in about an hour.

- 3 years, 6 months ago

Yeah , fine .

- 3 years, 6 months ago

Done it.

- 3 years, 6 months ago

Thanks :) But it seems there's an extra dot that's come out of nowhere ;)

- 3 years, 6 months ago

Yes. Just a editing mistake.

- 3 years, 6 months ago

Ok .

- 3 years, 6 months ago

- 3 years, 6 months ago

All the best dear. You're making it Great. Keep it up. @Rajdeep Dhingra

- 3 years, 6 months ago

Thank you sir ! You always motivate people. Proud to be your student.

- 3 years, 6 months ago

You noticed it too ! That Megh was online today .

- 3 years, 6 months ago

Yes

- 3 years, 6 months ago

I am also participating. @Azhaghu Roopesh M Let's see. :) :) :)

- 3 years, 6 months ago

Challenge accepted :P

- 3 years, 6 months ago

Yes . Though I'll submit my entries after the 4th .

- 3 years, 6 months ago

No Problem. Best of luck.

- 3 years, 6 months ago

Thanks :)

- 3 years, 6 months ago