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

An equilateral triangle

KKK

Let P be any point inside of an equilateral triangle \Delta ABC. Let D, E, F be the projection from P to BC, CA, AB. Let A', B', C' are the projections from A, B, C to the line pass through P.

Prove that PE\times BB'+PF\times CC'=PD\times AA'

Author: Van Khea, Prey Veng, Cambodia

Geometry, Training

2017

let D, E, F be midpoints of BC, CA, AB. Let (d) be any line and let AA_1, DD_1, EE_1, FF_1 be distances from A, D, E, F to the line (d).

Prove that  \overline{AA_1}=\overline{DD_1}+\overline{EE_1}+\overline{FF_1}

Author: Van Khea, Prey Veng, Cambodia

 

Training Olympiad

Let A_1, A_2, ..., A_n lies on circle (O) such that A_1A_2=A_2A_3=...=A_nA_1. Prove that:

\displaystyle \frac{1}{A_1A_2^2}+\frac{1}{A_1A_3^2}+...+\frac{1}{A_1A_n^2}=\frac{n^2-1}{12R^2}, \forall{n\ge 3}.

Author: VanKhea

Generalization: RagvaloD

ទ្រឹស្ដីបទមេនេឡុស និងអនុវត្ត

ME

Let P, Q, R be collinears points on the sides BC, CA, AB.

Prove that if  \displaystyle \frac{AR}{AB}-\frac{BP}{BC}+\frac{CQ}{CA}=1 then \displaystyle \frac{AB}{AR}-\frac{BC}{BP}+\frac{CA}{CQ}=-2

Author: Van Khea