# 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)