A Geometry Proof Problem

Hi, I have been stuck on this problem, please help me.

Problem: \textbf{Problem:}

Let M M be an arbitrary point lying outside of parallelogram ABCD ABCD . Consider line l1 l_1 passing through A A parallel to MC MC , line l2 l_2 passing through B B parallel to MD MD , line l3 l_3 passing through C C parallel to MA MA and line l4 l_4 passing through D D and parallel to MB MB . Prove that all 4 4 lines l1,l2,l3,l4 l_1, l_2, l_3, l_4 share a common point.

Thoughts: \textbf{Thoughts:}

I am lost in how to really complete the proof of this problem, but I did notice that there are many more parallelograms in this picture. I'm not sure if this is helpful in any way, but it would be awesome if anyone has any other ideas.

Note by Matthew Kendall
4 years ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. 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"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Strategy: Define l1l2=Nl_1\cap l_2=N, prove that NN lies on the other two lines. btw: there's a form of symmetry happening here, what is it?

Xuming Liang - 4 years ago

Log in to reply

I found it, there is a reflection of two congruent triangles through the center of the parallelogram. Once you prove that, the other two lines definitely would intersect at N N . Thank you for the hint :)

Matthew Kendall - 4 years ago

Log in to reply

There are a lot of parallel lines, you should try by looking for angles

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...