My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum

Thanks Tree1Thanks
  • 1 Post By lemma28
LinkBack Thread Tools Display Modes
January 27th, 2013, 03:53 AM   #1
Joined: Jan 2013

Posts: 1
Thanks: 1

Difficult problem: deltoid inscribed into an ellipse

I'm a math teacher and I've found a very hard problem in one of my math classrooms' textbooks. It was firstly proposed as problem n. 9, back in 1995, in the "Annual Iowa Collegiate Mathematics Competition". Link is (no solution file available in the site for that year).

The text is rather simple:

Let E be an ellipse in the plane and let A be a fixed point inside of E. Suppose that two perpendicular lines through A intersect E in points P, P' and Q, Q' respectively. Prove that

is independent of the choice of lines.

Practically there's a "deltoid" (a quadrilateral with perpendicular diagonals) inscribed into a generic ellipse and the property to demonstrate involve the four segments in which the diagonals are reciprocally subdivided.

Till now I've found that:
  • the property is actually true. I've seen that by empirical verification using Geogebra and following the cartesian analytic solution (see (1) ) outlined below;[/*:m:2nbm9j8n]
  • using a cartesian reference system it's possible to prove the property, but with huge calculations (see (1) );[/*:m:2nbm9j8n]
  • drawing the tangent lines to the ellipse in the points P, P', Q, Q' they intersect in four points (say B, C, D, F). The diagonals of this new quadrilateral (this time circumscribed to the ellipse) meets too in the point A (!) (empirical but unproven discover using Geogebra - probably, the core of the yet unfound geometrical demonstration is connected to this fact).[/*:m:2nbm9j8n]

Since I'm not satisfied with the analytical cartesian way (with huge calculations) and I don't think that was the proof intended by the people that proposed this problem back in 1995 and since I think there must be a clever and a more elegant geometrical way to it, I'm asking for some help to find a better solution.
I've also challenged my math students (aged 16-18, about 70 students) to find a solution to this problem, and I'm curious to see if a fresher (and just less experienced) brain than mine can find the right way when my purported experience isn't much helpful in this case.

(1) Given the ellipse centered in the origin with equation , a generic point inside the ellipse (that is with ) and the two perpendicular lines r: and s: incident in A that intersect the ellipse respectively in the points P, P and Q, Q', after calculating the coordinates of the intersection points P, P', Q and Q' of r and s with the ellipse and the lengths of the segments AP, AP', AQ, AQ', then it is


so that
and this last expression doesn't actually depend on the coefficient m, that is on the choice of the two lines r and s.
Thanks from Greek3
lemma28 is offline  
August 7th, 2014, 08:00 AM   #2
Joined: Aug 2014
From: Milan

Posts: 2
Thanks: 0

Could you write all the passages that led you to the solution???

Thanks a lot!!!
Greek3 is offline  
August 10th, 2014, 10:35 AM   #3
Global Moderator
Joined: Dec 2006

Posts: 20,965
Thanks: 2214

This is more readable.
skipjack is offline  

  My Math Forum > High School Math Forum > Algebra

deltoid, difficult, ellipse, inscribed, problem

Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
inscribed circle center problem elim Algebra 5 January 22nd, 2011 07:36 AM
another ellipse problem smash Algebra 13 January 12th, 2011 08:25 PM
Problem with an ellipse peachy1087 Algebra 1 June 15th, 2010 07:43 PM
Help on Circum/inscribed Hexagon Problem Hangman Algebra 3 April 8th, 2009 04:28 AM
Problem with ellipse Grigory_Perelman Algebra 9 December 20th, 2006 02:53 PM

Copyright © 2019 My Math Forum. All rights reserved.