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Menelaus's Theorem

Date: 01/25/99 at 11:54:49
From: Wanwipa
Subject: Using Menelaus' theorem to prove this problem

I want to know how to prove this problem by using Menelaus' theorem:

A straight line intersects sides AB, BC and the extension of side AC of 
a triangle ABC at points D, E and F respectively. Prove that the 
midpoints of the line segments DC, AE and BF lies on a straight line.

Thank you

Date: 01/26/99 at 09:27:17
From: Doctor Floor
Subject: Re: Using Menelaus' theorem to prove this problem

Hi Wanwipa,

Thanks for your question.

First, for your reference, here is a diagram of the set-up:

Note that in this picture, points D, E and F are not the same as in 
your question. Gauss' theorem (1810) says: The midpoints of segments 
AD, BE and CF are collinear. In the figure they are marked G, H and I.

Now apply Menelaus' theorem on triangle ABC and line DE, to find:

  AF   BE   CD
  -- * -- * -- = -1
  FB   EC   DA

Consider triangle JKL, made out of the side midpoints of triangle ABC.

You can see that JI = 0.5 BE and IL = 0.5 EC, so BE/EC = JI/IL. In the 
same way you find: AF/FB = LH/HK and CD/DA = KG/GJ. So we have:

 LH   JI   KG
 -- * -- * -- = -1
 HK   IL   GJ

So, again applying Menelaus' theorem, this must mean that G, H, and I 
are collinear. That is the Gaussian line.

If you have a math question again, please send it to Dr. Math. If 
necessary, don't hesitate to write back.

Best regards,

- Doctor Floor, The Math Forum   
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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