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        riddles >> medium >> Dijkstra, AMM and geometry
        
(Message started by: Aryabhatta on Apr 19^th, 2007, 10:58am)

Title: Dijkstra, AMM and geometry
Post by Aryabhatta on Apr 19^th, 2007, 10:58am

A geometry problem which interested Dijkstra, from AMM 1998.

(AMM = American Mathematics Monthly)

In the figure below,

If AD + DF equals AB + BF

Show that

AE + EF = AC + CF

Title: Re: Dijkstra, AMM and geometry
Post by rmsgrey on Apr 20^th, 2007, 8:26am

One observation which may lead nowhere: A and F are focii of an ellipse, and B and D points on the perimeter.

Title: Re: Dijkstra, AMM and geometry
Post by Barukh on Apr 27^th, 2007, 9:51am

Here’s a nice solution I learned elsewhere.

If we can show that, under given conditions, there exists a circle touching four lines AC, AD, CD, BE (prolonged, if needed – see the drawing), that the statement of the problem will follow (why?)

To show that such a circle indeed exists, consider the ex-circles of triangles ACD and ABE (the ex-circle of a triangle is a circle tangent to 2 of its sides internally and one externally).

The center O₁ of the ex-circle of ACD is the intersection of angle bisectors A and CDE.
The center O₂ of the ex-circle of ABE is the intersection of angle bisectors A and CBE.

Now, consider the triangle FF₁F₂, where F₁, F₂ are chosen on corresponding lines such that BF₁ = BF, DF₂ = DF. Since AF₁ = AB+BF₁ = AB+BF = AD+DF = AD+DF₂ = AF₂, we get that all three triangles F₁AF₂, F₁BF, F₂DF are isosceles.

Therefore, the bisectors of angles A, CBE, CDE coincide with the bisectors of corresponding segments F₁F₂, FF₁, FF₂. But the latter 3 bisectors are concurrent, since they define the center of the circumcircle of the triangle F₁FF₂. Thus, O₁ and O₁ coincide.

Finally, the ex-circles of triangles ACD and ABE also coincide, since they touch the same two lines and are centered at the same point. But now it follows that this circle touches all four needed lines!

Title: Re: Dijkstra, AMM and geometry
Post by Aryabhatta on Apr 27^th, 2007, 12:49pm

Interesting solution Barukh!

Title: Re: Dijkstra, AMM and geometry
Post by Barukh on Apr 27^th, 2007, 11:10pm

Thanks. Some interesting results about this theorem:

1. It is attributed to Australian mathematician M. L. Urguhart who discovered the result in 1960s while "considering some of the fundamental concepts of the theory of special relativity" (!)

2. The result was proved as early as 1841 by English mathematician De Morgan.

3. The theorem has the following nice equivalent statement (partly spotted by rsmgrey): If C and E are points on an ellipse with foci A and F, then points of intersection AC, EF and AE, CF lie on a confocal ellipse".