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

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Open Proof Contest 3

Cancelled\huge \textbf{Cancelled}Cancelled

Comments

$\huge \textbf{Cancelled}$