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

Polygone identity

2017

Let A_1, A_2, ..., A_k, k\ge 3 lies on circle  (O) such that A_1A_2=A_2A_3=...=A_kA_1 and let P\in (O) such that arc PA_1=m.arc PA_2. Prove that:

\displaystyle \frac{1}{PA_1^2}+\frac{1}{PA_2^2}+...+\frac{1}{PA_k^2}=\frac{k^2}{4R^2sin^2(m\pi)}

Author: Van Khea

Generalization: Tintarn

Geometry for Test

mm

Given a triangle \Delta ABC such that AC=\sqrt{2}R and \angle B=45^0.

Prove that \displaystyle \frac{1}{BA}+\frac{1}{BC}=\frac{\sqrt{2+\sqrt{2}}}{R}

Author: Van Khea

Centroid G of ABC: Author Van Khea

G

Let G be the centroid of \Delta ABC. Let AG, BG, CG cuts BC, CA, AB at D, E, F and cuts circumcircle of \Delta ABC at X, Y, Z. Let YZ, ZX, XY cuts $AG, BG, CG$ at P, Q, R.

a) Prove that: \displaystyle \frac{GD}{GX}+\frac{GE}{GY}+\frac{GF}{GZ}=\frac{3}{2}

b) Prove that: \displaystyle \frac{GD}{GP}+\frac{GE}{GQ}+\frac{GF}{GR}=3

Author: Van Khea

author by vankhea

NN

Let P be any point inside of \Delta ABC. Let AP, BP, CP cuts BC, CA, AB at D, E, F and cuts circumcircle of \Delta ABC at X, Y, Z.

Prove that: \displaystyle (\frac{XP}{PA}+\frac{YP}{PB}+\frac{ZP}{PC})(\frac{AD}{DX}+\frac{BE}{EY}+\frac{CF}{FZ})\geq 27

Author: Van Khea