Παρασκευή, 4 Νοεμβρίου 2016

Harmonic Mean in Geometry Ο αρμονικός μέσος στη γεωμετρία !

Harmonic Mean in Geometry



Harmonic Mean in Geometry

The harmonic mean H(a,b) of two positive numbers a and b is defined as

H(a,b)=21a+1b,
and equivalently

2H(a,b)=1a+1b,
or else

H(a,b)=2aba+b.
It makes a most dramatic appearance in the problem of average speed, but also pops up in various geometric circumstances. Below I collect a few examples.


  1. Let AGB be a semicircle and AC=a, BC=b.




    If CGAB and CHOG then GH=H(a,b).




  2. In a trapezoid ABCD, let EF be the line parallel to the bases AB and CD through the point of intersection of the diagonals AC and BD:




    If CD=a and AB=b, then EF=H(a,b).



    It is worth observing that the point of intersection of the diagonals divides EF into two equal parts, which leads to another common example:


  3. In a 1877 sangaku from the Hyogo prefecture, AE and BD are both perpendicular to AB. P is the intersection of AD and BE, and CPAB.




    If AE=a and BD=b, then CP=2H(a,b).





  4. The same motif reemerges in the See-Saw Lemma where AE, BF, and EF are tangent (the latter at X) to the semicircle AXB.




    If AE=a and BF=b, then XY=H(a,b).




  5. Vladimir Nikolin from Serbia has observed appearance of the harmonic mean in his Rhombus Lemma.




    Let AD be the bisector of A in ΔABC, BD||AC and CD||AB. Then ABDC is a rhombus and its side p is half of the harmonic mean of sides b=AC and b=AB.



    Choosing the angle at A to be right leads to a special case where the rhombus becomes a square.




  6. Let a square be inscribed into a right triangle as in the diagram




    Then, if a and b are the legs of the triangle and p the side of the square, 1p=1a+1b. In other words, p=12H(a,b).




  7. As a consequence of the Rhombus Lemma there is a simple way of constructing the harmonic ratio:




    In ΔABC, with AB=c and AC=b, draw AD - the bisector of A. At D erect a perpendicular to AD and let E be its intersection with either AB or AC. Then AE=H(b,c).



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