AoPS - Problem

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Problem

Source: Bosnia and Herzegovina TST 2010 problem 2

Tags: ratio, geometry proposed, geometry

gobathegreat

30.03.2014 17:59

Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$ . Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$ . Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$ .

Number1

30.03.2014 20:45

Projective solution: Cross ratio $\begin{eqnarray*} (A,B;Q,R) &=& (PA,PB;PQ,PR) \\ &=& (PA,PB;PF,PD) \\ &=& (A,B;F,D) = constant = c\ \end{eqnarray*}$ so ${AQ\over QB} : {AR\over RB} = c\;\;\; (*)$ Let $AQ= x, QR =y, RB=z$ and $AB= a$ . So we have $a=x+y+z$ and from (*) $xz = c(ay+zx)$ . So ${AQ\cdot RB\over QR} = {xz\over y} = {ac\over 1-c} = constant$ .