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A Problem in the Geometry of Triangles


A problem in the geometry of triangles

ABC
 Isosceles triangle has its head A Where the angle ABC = 2
π/5

and k is a relative integer , we put BC=a .   

Bisector of the angle ABC cut the line segment [AC] In point D.

[DH],[DI],[AJ] and [BK]  Are the elevations in the triangles DAB , DBC , ABC , DBC  Respectively.

1. complete the shape.

2.

    a. Prove that the two triangles DAB and DBC are Isosceles.

   b.Consider the triangle,  Express length AB a function of cos(π/5) and a , Infer the phrase CD a function of cos(π/5) and a

   c. Express the triangle DBC Proved that, CD=2a cos(2π/5)

   d. Infer  that  cos(π/5)-cos(2π/5)=1/2

3.  

    a. Express the triangle DBC ,  Express length IB a function of cos(π/5) and a ,  Infer the phrase CD a function of cos(π/5) and a

       b. Prove that IC= 2a [cos²(2π/5)]

      c. Infer  that  cos(π/5)+cos²(2π/5)=1

4.

    a. Solve in ℝ the following equation:  x²+1/2 x-1/4 

     b. Check that cos(π/5) is a solution to this equation

    c.  Prove that for every real number x it is: cos(sinx)>sin(cosx)



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