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Puzzling Geometry
Posted:
Nov 9, 2010 10:17 AM
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In my previous 2 posts, I had mentioned about a paper by Tom Rike.In this paper he has mentioned following problem proposed by Sergey Yurin-
In an isosceles triangle ABC, AB =AC, and angle=20 degree.Point P is taken on the side AC such that AP=BC.Find angle PBC.
I had also proposed same problem in May 1997.I was inspired by following problem, which is also
very famous problem-
In an isosceles triangle ABC with AB=AC and vertical angle 100 degree , the side AC is extended
up to a point D such that AD=BC.Find angle CBD.
Again in May 2002, I came to following problem-
Let ABC be a triangle in which AB=AC and angle BAC=140 degree .Extend CA up to K such that
AK=BC.Find angle BKA. A lies between K and C.
Here all 3 triangles are isosceles with vertical angles 20, 100 and 140 degrees.If you do some calculations , you will find these triangles are interconnected with equation (x^3) ? 3*x + 1=0.
Other than this, these triangles have beautiful relationship between sides and angle bisector in
the following way-
Let ABC be triangle with angle BAC=20 degree and AB=AC .Suppose the internal bisector of
angle B meets AC in K.Then we will have BC=AK-BK.
Now for another triangle-
Consider a triangle ABC in which AB=AC and angle BAC=100 degree .Let the internal bisector
of angle ABC meet AC in K.Then we have BC=BK+KA.
Now for the 3rd triangle-
Let ABC be an isosceles triangle with AB=AC and angle BAC=140 degree .Suppose the external
bisector of angle B intersects CA in D.Then , BC=AD+BD.
Now , we can find that one of the roots of above mentioned equation (x^3) ? 3*x + 1=0
is 2*sin (pi/18).
We can express sides of above 3 isosceles in terms of 1, 2*sin(pi/18) and 1 ? 2*sin(pi/18).
This number 2*sin(pi/18) is somewhat as interesting as Golden Ratio.
In one of the books about works of Sreenivasa Ramanujan, I find 2*sin(pi/18) can be expressed
in terms of nested radicals in following way
2*sin(pi/18)=(2-(2+(2+(2-....)^0.5)^0.5)^0.5)^0.5
Other than this 2*sin(pi/18) is Bond Percolation Threshold for Triangular Lattice and 1-2*sin(pi/18)
is Bond Percolation Threshold for Honeycomb lattice.
I had got these things published with some more problems in January 2003 issue of Samasya
with title ?Adventitious Angles in Angle Chasing?.In fact I had posted these things from July 2002 to
May 2003 on this site only.I am posting it again so that if some one have not come across these
things can try to prove it and can feel its fun.One will reel under real joy and pleasure, once one
discovers the fascinating , adventitious and chaotic nature of Euclidean Geometry.The universe
of Euclidean Geometry makes us wonder the way the new realations unfold themselves.
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