3 centers collinear


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


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


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



Let circle (C) inscribe in square ABCD. Let P be point lie on circle (C).

Prove that PA^6+PB^6+PC^6+PD^6=252.r^6

Author: VanKhea, Cambodia

The distance from a point to the side of an equilateral triangle


Let I be incenter of an equilateral triangle \Delta ABC. Let P be any point and let D, E, F be projection points from P to the sides BC, CA, AB.

Prove that \displaystyle PD^2+PE^2+PF^2=3r^2+\frac{3}{2}.PI^2

Author: Van Khea, Cambodia

Equality in Triangle


Let O be circumcenter of an equilateral triangle \Delta ABC and let P be any point in the plan.

Prove that PA^4+PB^4+PC^4=3R^4+3PO^4+12.PO^2.R^2

Author: Van Khea, Cambodia