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