## USAMO 2009/5

### USAMO 2009/5

Trapezoid $ABCD,$ with $\overline{AB}||\overline{CD}$, is inscribed in circle $\omega$ and point $G$ lies inside triangle $BCD$. Rays $AG$ and $BG$ meet $\omega$ again at points $P$ and $Q$, respectively. Let the line through $G$ parallel to $\overline{AB}$ intersect $\overline{BD}$ and $\overline{BC}$ at points $R$ and $S$, respectively. Prove that quadrilateral $PQRS$ is cyclic if and only if $\overline{BG}$ bisects $\angle CBD$.

"Questions we can't answer are far better than answers we can't question"

### Re: USAMO 2009/5

i have solved this . but my solution is too large

main part of my solution is to prove $DX$=$CY$ where $X=QR\cap w ; Y=PS\cap w$

main part of my solution is to prove $DX$=$CY$ where $X=QR\cap w ; Y=PS\cap w$

### Re: USAMO 2009/5

$\text{My proof}$:

$\text{If part}$:

Extend $QR$ to $T$.Join $B,T$.$BTRG$ is cyclic.We can also see that $PCSG$ is cyclic.We can prove that $T,G,C$ are collinear.$\therefore$ $\angle RQP+\angle PSR=180^{\circ}-\angle TBP+\angle PCT=180^{\circ}-\angle PCT+\angle PCT=180^{\circ}$.

$\text{Only if part}$:

The steps can be reversed.

$\text{If part}$:

Extend $QR$ to $T$.Join $B,T$.$BTRG$ is cyclic.We can also see that $PCSG$ is cyclic.We can prove that $T,G,C$ are collinear.$\therefore$ $\angle RQP+\angle PSR=180^{\circ}-\angle TBP+\angle PCT=180^{\circ}-\angle PCT+\angle PCT=180^{\circ}$.

$\text{Only if part}$:

The steps can be reversed.

"Questions we can't answer are far better than answers we can't question"

### Re: USAMO 2009/5

Previously posted in IMO Marathon, you can see a few more solutions there -

viewtopic.php?p=13226#p13226

viewtopic.php?p=13226#p13226

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Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

### Re: USAMO 2009/5

Sorry for double posting.Actually I have searched but nothing was found.*Mahi* wrote:Previously posted in IMO Marathon, you can see a few more solutions there -

viewtopic.php?p=13226#p13226

"Questions we can't answer are far better than answers we can't question"