CTK Exchange
Front Page
Movie shortcuts
Personal info
Awards
Reciprocal links
Privacy Policy

Interactive Activities

Cut The Knot!
MSET99 Talk
Games & Puzzles
Arithmetic/Algebra
Geometry
Probability
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
My Logo
Math Poll
Other Math sit's
Guest book
News sit's

Manifesto: what CTK is about |Store| Search CTK Buying a book is a commitment to learning Table of content Things you can find on CTK Chronology of updates Email to Cut The Knot

CTK Exchange

Subject: "geometry"     Previous Topic | Next Topic
Printer-friendly copy     Email this topic to a friend    
Conferences The CTK Exchange High school Topic #204
Reading Topic #204
golland
Member since Apr-30-02
Oct-02-02, 09:36 AM (EST)
Click to EMail golland Click to send private message to golland Click to add this user to your buddy list  
"geometry"
 
  
A triangle and a point outside of it. Draw a line through the point
that will divide triangle into 2 equal areas.

BG


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
NJZ
Member since Mar-23-02
Oct-04-02, 06:30 AM (EST)
Click to EMail NJZ Click to send private message to NJZ Click to view user profileClick to add this user to your buddy list  
1. "RE: geometry"
In response to message #0
 
   Given an arbitrary triangle (defined by vertices at points A, B, C) and a point P outside of the triangle, first draw two lines from P to the two vertices (A, B) such that neither line intersects the triangle, and the sum of their Euclidean distances is the maximum possible, given the relative locations of ABC and P in the plane. Next draw a line through P such that it bisects he angle APB. This line also divides the area of triangle ABC in half.

I didn’t actually prove this method, but it appears to be empirically and intuitively correct. I believe it could be proved (or disproved) using a system of linear equations to define the lines forming the triangle and the bisecting line, and their corresponding integrals to measure the relevant areas. Please let me know if this is correct, thanks!

NJZ


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
NJZ
guest
Oct-04-02, 09:56 AM (EST)
 
2. "RE: geometry"
In response to message #1
 
  
Hi NJZ,

Thanks for the reply.

In your solution the position of the third vertex C is irrelevant.
But the result will change significantly with change in C relative
to points A and B.


Golland.


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
NJZ
Member since Mar-23-02
Oct-04-02, 11:18 AM (EST)
Click to EMail NJZ Click to send private message to NJZ Click to view user profileClick to add this user to your buddy list  
3. "RE: geometry"
In response to message #2
 
   You are absolutely right. Obviously, my solution only works in special cases. I should have given this one a bit more thought before posting my answer. I'll have work on this one some more, and see if I can come up with a correct general solution.

NJZ


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top

Conferences | Forums | Topics | Previous Topic | Next Topic

You may be curious to visit the old CTK Exchange archive.

|Front page| |Contents| |Store|

Copyright © 1996-2018 Alexander Bogomolny

[an error occurred while processing this directive]
 Advertise