318. IRAN - 2011 (geometrie).

by Virgil Nicula, Sep 17, 2011, 3:06 AM

An interesting metrical remarks.

IRAN, 2011. Let $\triangle ABC$ with $b\ne c$ , the incircle $w=C(I,r)$ and $\left\{\begin{array}{cc} X\in BC\cap w\ ; & Y\in CA\cap w\\\\ Z\in AB\cap w\ ; & L\in BC\cap YZ\end{array}\right\|$ . Denote the simmedian center $S$ of $\triangle ABC$ and $L\in BC\cap AA$ .

Prove that $L\in YZ\iff$ $L\in OI\iff $ $IS\parallel BC\iff$ $\Gamma N\parallel BC\iff$ $IO\perp AX\iff$ $a(b+c)=b^2+c^2\iff $ $(s-a)^2=(s-b)(s-c)$ , where $2s=a+b+c$ .

Proof.

$\blacktriangleright\ \boxed{L\in YZ}\iff$ $\frac {LB}{LC}=\frac{XB}{XC}\iff$ $\frac {c^2}{b^2}=\frac {s-b}{s-c}\iff$ $s\left(b^2-c^2\right)=b^3-c^3\iff$ $(a+b+c)(b+c)=2\left(b^2+bc+c^2\right)\iff$ $\boxed{a(b+c)=b^2+c^2}$ .

$\blacktriangleright\ \boxed{IS\parallel BC}\iff$ $[BSC]=[BIC]\iff$ $\frac {a^2}{a^2+b^2+c^2}=\frac {a}{2s}\iff$ $a^2(a+b+c)=a\left(a^2+b^2+c^2\right)\iff$ $\boxed{a(b+c)=b^2+c^2}$

$\blacktriangleright\ \boxed{\Gamma N\parallel BC}\iff$ $\frac {a}{s-a}=\frac {a(s-a)}{(s-b)(s-c)}\iff (s-a)^2=(s-b)(s-c)$ , where $N$ is the Nagel's point and $\Gamma$ is the Gergonne's point.

$\blacktriangleright\ \boxed{(s-a)^2=(s-b)(s-c)}\iff$ $a^2-2a(b+c)+(b+c)^2=a^2-(b-c)^2\iff$ $\boxed{a(b+c)=b^2+c^2}$ .

$\blacktriangleright\ \boxed{IO\perp AX}\iff$ $OA^2-OX^2=IA^2-IX^2\iff$ $XB\cdot XC=IA^2-IX^2\iff$ $\boxed{(s-b)(s-c)=(s-a)^2}$ .

$\blacktriangleright\ \boxed{L\in YZ}\iff$ $X$ is the conjugate of $L$ w.r.t. $\{B,C\}$ $\iff$ $AX$ is polar line of the point $L \iff$ $LO\perp AX$ $\iff \boxed{L\in OI}$ .

Remark. Let $P(x,y,z)\ ,\ R(u,v,w)$ be two points with the mentioned barycentrical coordinates and which belong to inside of $\triangle ABC$ . Denote $\left\{\begin{array}{c} D\in BC\cap AP\\\\ E\in BC\cap AR\end{array}\right\|$ .

Then, using the van Aubel's theorem, obtain that $PR\parallel BC\iff \frac {PA}{PD}=\frac {RA}{RE}\iff$ $\frac {y+z}x=\frac {v+w}u$ . In conclusion, $\boxed{\ PR\parallel BC\iff u(y+z)=x(v+w)\ }$ .
This post has been edited 14 times. Last edited by Virgil Nicula, Nov 20, 2015, 6:04 AM

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