Easy Problem in geo, (cyclic, angle chasing)
- Kazi_Zareer
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$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$. (Source:china TST 2014 P1)
We cannot solve our problems with the same thinking we used when we create them.
- nahin munkar
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Re: Easy Problem in geo, (cyclic, angle chasing)
Here,we can show $AQYR$ & $XQDP$ is cyclic. So,By easy angle chasing, we can show $\angle RQF$ =$\angle FQP$.That's the proof.[here, RG cuts AC at Y & EP cuts BD at X. It's a simple proof.Try to yourself.]. Proved.
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss
Re: Easy Problem in geo, (cyclic, angle chasing)
Apply a projective transformation that sends $ABCD$ to a square $A_{1}B_{1}C_{1}D_{1}$.The rest is very easy
"Questions we can't answer are far better than answers we can't question"
- nahin munkar
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Re: Easy Problem in geo, (cyclic, angle chasing)
Well done.It's another good approach in projective way.
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss
- Kazi_Zareer
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- Joined:Thu Aug 20, 2015 7:11 pm
- Location:Malibagh,Dhaka-1217
Re: Easy Problem in geo, (cyclic, angle chasing)
I have already solved it and given the main clues of my solution.nahin munkar wrote:Here,we can show $AQYR$ & $XQDP$ is cyclic. So,By easy angle chasing, we can show $\angle RQF$ =$\angle FQP$.That's the proof.[here, RG cuts AC at Y & EP cuts BD at X. It's a simple proof.Try to yourself.]. Proved.
We cannot solve our problems with the same thinking we used when we create them.
- Kazi_Zareer
- Posts:86
- Joined:Thu Aug 20, 2015 7:11 pm
- Location:Malibagh,Dhaka-1217
Re: Easy Problem in geo, (cyclic, angle chasing)
Awesome sala!tanmoy wrote:Apply a projective transformation that sends $ABCD$ to a square $A_{1}B_{1}C_{1}D_{1}$.The rest is very easy
We cannot solve our problems with the same thinking we used when we create them.
- nahin munkar
- Posts:81
- Joined:Mon Aug 17, 2015 6:51 pm
- Location:banasree,dhaka
Re: Easy Problem in geo, (cyclic, angle chasing)
Yes,dude.I solved from those 2 clues-(cyclic,angle-chasing).Kazi_Zareer wrote:I have already solved it and given the main clues of my solution.nahin munkar wrote:Here,we can show $AQYR$ & $XQDP$ is cyclic. So,By easy angle chasing, we can show $\angle RQF$ =$\angle FQP$.That's the proof.[here, RG cuts AC at Y & EP cuts BD at X. It's a simple proof.Try to yourself.]. Proved.
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss