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

Right triangle inequality

Let \Delta ABC be right triangle which \angle A=90^0, BC=a, CA=b, AB=c.

Prove that a^3+b^3+c^3\ge (2+\sqrt{2}).abc.

Author: Van Khea, Cambodia

A problem of square

ss

Let ABCD be square and let X, Y, Z, T be midpoints. For any point P inside of square ABCD, Prove that:

\sum PA^2-\sum PX^2=a^2. ( a=AB )

Author: Van Khea, Cambodia