An equilateral triangle ABC

2017

Let O be circumcenter of an equilateral triangle \Delta ABC and let P be point lie on incircle. Prove that:

a) PD=PE+PF

b) PD^2+PE^2+PF^2=3R^2

c) PD^4+PE^4+PF^4=\frac{9}{2}.R^4

Author: Van Khea, Cambodia

A beautiful problem

17

Let P be point lie on circumcircle of an equilateral triangle \Delta ABC. Let (D_1D_2)//(BC) ;(E_1E_2)//(CA); (F_1F_2)// (AB)

Prove that D_1D_2+E_1E_2+F_1F_2=2.AB

Author: Van Khea, Cambodia

Geometry, Equilateral triangle

17

Let P be point lie on circumcircle of an equilateral triangle \Delta ABC.

Let D, E, F be projection points from P to BC, CA, AB.

a) Prove that \displaystyle AD^2+BE^2+CF^2=\frac{33}{4}.R^2

b) Prove that \displaystyle AD^4+BE^4+CF^4=\frac{369}{16}.R^4

Author: Van Khea, Cambodia