An Equilateral problems

p2

Let D, E, F be midpoints of an equilateral triangle \Delta ABC. Let K be midpoint of EF. Prove that:

a) \displaystyle PD^2+PE^2=\frac{32}{3}.PF^2

b) \displaystyle PD^4+PE^4=\frac{530}{9}.PF^4

Author: Van Khea, Cambodia

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Geometry, triangle

p1

\Delta ABC is an equilateral triangle.

E, F is midpoints of CA, AB and CP\perp EF

Prove that \displaystyle sin^2(EFP)+sin^2(EPF)=\frac{1}{7}

Proposed by Van Khea, Cambodia

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

3 centers collinear

P1

Let D be point lie on BC of \Delta ABC. Let I and J be incenters of \Delta ABD and ACD. Let P, Q, R be circumcenters of \Delta AIJ, \Delta BIJ, \Delta CIJ.

Prove that P, Q, R are collinear.

Author: Van Khea, Cambodia

An equilater triangle

P2

Given an equilateral triangle \Delta ABC. Let D be point on [BC]. Let I and J be incenter of \Delta ABD and \Delta ACD. Let K be circumcenter of \Delta AIJ.

Prove that \Delta IJK is an equilateral triangle.

Author: Van Khea, Cambodia