A problem of square

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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

Square

2017

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

17

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

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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

some of problem in equilateral triangle

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Given an equilateral triangle \Delta ABC. Let (l) is a line tangent to incircle (I). Let P, Q, R be projection points from A, B, C to line (l).

Prove that IP^4+IQ^4+IR^4=33r^4

Author: Van Khea, Cambodia

Excircle

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Given an equilateral triangle \Delta ABC. Let P be any point lie on incircle.

Prove that PA^4+PB^4+PC^4=99r^4

Author: Van Khea, Cambodia

The equality in equilateral triangle ABC

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\Delta ABC is an equilateral triangle. P lie on circumcircle and Q lie on incircle. m, n, p are distances from P and x, y, z are distances from Q to BC, CA, AB.

Prove that \displaystyle \frac{m^2+n^2+p^2}{x^2+y^2+z^2}=2

Author: Van Khea, Prey Veng, Cambodia