Given a circle $\Gamma$ with center $O$ and radius $r$.let $AB$ be a fixed diameter of $\Gamma$. $K$ be a fixed point of segment $AO$. Denote by $t$ the line tangent to $\Gamma$ at $A$. For any chord $CD$ (other than $AB$) passing through $K$. Let $P$ and $Q$ be the points of intersection of lines $BC$ and $BD$ with $t$.
prove that the product $AP\cdot AQ$ remains costant as the chord $CD$ varies.
( dont give too stress on the line that THE CHORD CD VARIES...just fix AB the diameter and chose any chord CD on your own accord).
very very very very very nice problem.100% pure( ) euclidian geometry
the finest problem of apmc
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Last edited by Moon on Mon Jan 17, 2011 7:35 pm, edited 2 times in total.
Reason: LaTeXed! Now it is readable.
Reason: LaTeXed! Now it is readable.
women of purity are for men of purity and hence men of purity are for women of purity - THE HOLY QURAN
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Re: the finest problem of apmc
i am totally astonish isnt there anybody to answer THIS ques??
i was quite eager when i was posting this prob.cuz i wanna see the ideas,but no response!!!
i was quite eager when i was posting this prob.cuz i wanna see the ideas,but no response!!!
women of purity are for men of purity and hence men of purity are for women of purity - THE HOLY QURAN
Re: the finest problem of apmc
When posting a geometry problem, you should attach a diagram (or at least write it legibly with LaTeX) to make it more attractive to users.
BTW off topic, why don't you introduce yourself here: viewforum.php?f=10
I have solved your problem. But I'll be back after I add a feature in the forum.
BTW off topic, why don't you introduce yourself here: viewforum.php?f=10
I have solved your problem. But I'll be back after I add a feature in the forum.
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: the finest problem of apmc
Let $C'D'$ be any other chord through $K$. Define $P',Q'$ the same way we did $P,Q$.the arrivals wrote:Given a circle $\Gamma$ with center $O$ and radius $r$.let $AB$ be a fixed diameter of $\Gamma$. $K$ be a fixed point of segment $AO$. Denote by $t$ the line tangent to $\Gamma$ at $A$. For any chord $CD$ (other than $AB$) passing through $K$. Let $P$ and $Q$ be the points of intersection of lines $BC$ and $BD$ with $t$.
Prove that the product $AP\cdot AQ$ remains costant as the chord $CD$ varies.
Lemma 1: $PCDQ (=\omega_1), P'C'D'Q'(=\omega_2)$ both are cyclic.
Proof: $\angle QPB=90^o-\angle APB=\angle CAB=\angle CDB$. So $PCDQ$ is cyclic. Analogously $P'C'D'Q'$ is also cyclic.
Lemma 2: $P'PCC'$ cyclic.
Proof: $\angle AP'B=\angle C'D'B=\angle C'CB$.
From lemma 2, we have $BC'\cdot BP'=BC\cdot BP$ (power of point). $B$ is on the radical axis of both $\omega_1,\omega_2$. Again from power of point $C'K\cdot KD'=CK \cdot KD'$. So $BK$ is the radical axis of $\omega_1,\omega_2$.
Let $\omega_1, \omega_2$ intersect at $X,Y$. Then from power of point $PA\cdot AQ=YA \cdot AX=P'A \cdot AQ'$.
Proved!
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
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Re: the finest problem of apmc
umm...you think complex.at least your diagram is enough to make any novice afraid of geometry.
well but nice solution indeed.
well but nice solution indeed.
women of purity are for men of purity and hence men of purity are for women of purity - THE HOLY QURAN