AoPS - Problem

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Problem

Source: Romanian JBMO TST 2005 - day 2, problem 3

Tags: geometry, algebra, polynomial

Valentin Vornicu

02.04.2005 15:18

Let $ABC$ be an equilateral triangle and $M$ be a point inside the triangle. We denote by $A'$ , $B'$ , $C'$ the projections of the point $M$ on the sides $BC$ , $CA$ and $AB$ respectively. Prove that the lines $AA'$ , $BB'$ and $CC'$ are concurrent if and only if $M$ belongs to an altitude of the triangle.

Myth

02.04.2005 16:53

I know it is for kids

Suppose wlog

$AB=1$ ,

$BA'=x$ ,

$CB'=y$ and

$AC'=z$ . Then using Carno's theorem we obtain

$x^2+y^2+z^2=(1-x)^2+(1-y)^2+(1-z)^2$ (1) and from Ceva's theorem

$xyz=(1-x)(1-y)(1-z)$ (2). It follows from (1) that

$2(x+y+z)=3$ , and using this equality we can write (2) as

$(1-2x)(1-2y)(1-2z)=0$ .