Two circles intersect at A and B. A line through A cuts the first circle at P and the second circle at Q. At P, a tangent PT is drawn and TQ produced meets the second circle again at R. Prove that the points P, T, R and B are concyclic.
α α္αိုα္းႏွα ္αု $ \displaystyle A$ αဲα $ \displaystyle B$ αွာ ျαα္αΎααါαα္။ $ \displaystyle A$ αို ျαα္αΏαီး α်α₯္းαα ္ေαΎαာα္း αြဲαာαွာ αααα α္αိုα္းαို $ \displaystyle P$ αွာ ျαα္αΏαီး αုαိα α α္αိုα္းαို $ \displaystyle Q$ αွာ ျαα္αါαα္။ α‘αွα္ $ \displaystyle P$ αွာ αα္းαိα်α₯္းαα ္ေαΎαာα္း $ \displaystyle PT$ αို αြဲαိုα္αΏαီး ααα္ α‘αွα္ $ \displaystyle T$ αွ $ \displaystyle TQ$ αို αα္αြဲαာ αုαိα α α္αိုα္းαို $ \displaystyle R$ αွာαα္αΏαီး ျαα္αါαα္။ $ \displaystyle P,T,R,B$ αိုαဲ့ α‘αွα္ေαးαုαာ α α္αိုα္းαα ္αုαဲေαααွာ αွိေαΎαာα္း αα္ေαျααါ။
| αီေαးαြα္းαွာ α α္αိုα္းႏွα ္αု αိုααဲေျαာαΏαီး α α္αိုα္းႏွα ္αုαာ α‘ျαားαα္ααα္α်α္ ααါαα္αါαူး။ αါ့ေαΎαာα့္ α α္αိုα္းႏွα ္αုαို αြဲαဲ့ α‘αါ α‘αြα္ααူ αα္αူααီαဲ့ α α္αိုα္းႏွα ္αု α‘ျαα ္αြဲαα့္αါαα္။ αα္αူααီαူး αိုααα္း ေျαာααားαဲ့ α‘αြα္ αα္αူαီαာ αြဲαα္ေαာ ααႏိုα္αူးαားαိုα ေαးα αာ αွိαါαα္။ αြဲαိုαα αႏိုα္αါαα္။ αါေααα့္ αα္αူαီαဲ့ αုα္ααၱိေαြαို αံုးαြα့္ααွိαါαူး။ αα္αူαီျαα္းေαΎαာα့္ (α) α‘α်α္းαα္αူ၊ (α) αα္αူαီေαးααိဳးα်ား၊ (α) αα္αူαီ α‘αα္းαိုα္းα်ား αိုαဲ့ αုα္ααၱိေαြαို α‘ျαα္α‘α αွားαြα္း α‘αံုးျαဳαိαα္αါαα္။ ေαးα်α္α‘α ααါαα္αဲ့ αုα္ααၱိ ေαြαို αံုးαြα့္ααွိαါαူး။ αါေαΎαာα့္ α‘αိုαုα္ααၱိေαြ ααါαα္ေα αဲ့ αံုα်ိဳးαိုαာ ေαးαြဲαα့္αါαα္။ |
ေαးαြα္းαါ ေαးα်α္α‘α ...
$ \displaystyle PT$ α $ \displaystyle \text{tangent}$ ျαα
္αΏαီး $ \displaystyle PA$ α αααα
α္αိုα္းαဲ့ ေαးααိဳးျαα
္αာေαΎαာα့္ $ \displaystyle \text{Theorem 4}$ αို αံုးႏိုα္αဲ့ α‘αြα့္α‘ေαး αွိαါαα္။
$ \displaystyle PTQ$ α ααိαံျαα ္αာေαΎαာα့္ α‘αြα္းေαာα့္α်ား ေαါα္းျαα္း $ \displaystyle 180{}^\circ $ ျαα ္αα္ αိုαဲ့ αုα္ααၱိ၊ ααိαံ၏ α‘ျαα္ေαာα့္αα္ α‘αြα္းα်α္ႏွာα်α္း ေαာα့္αα ္α ံု ေαါα္းျαα္းαဲα αီαα္αိုαဲ့ αုα္ααၱိα်ားαို αံုးႏိုα္αဲ့ α‘αြα့္α‘ေαး αွိαါαα္။
$ \displaystyle P,T,R,B$ αို $ \displaystyle \text{concyclic}$ ျαα
္ေαΎαာα္း αα္ေαျααိုα $ \displaystyle P,T,R,B$ α‘αွα္ ေαးαွα္αို αα္αြα္αΏαီး α
αုαံ၏ α‘αြα္းေαာα့္ αα
္α
ံုေαါα္းျαα္း $ \displaystyle 180{}^\circ $ ျαα
္αွ်α္ ေαာα့္α
ြα္းαွα္α်ား α
α္αိုα္းေαααြα္ α်ေαာα္αα္ (αα
္α
α္αα္းαဲ αွိαα္) αိုαဲ့ $ \displaystyle \text{Theorem 9}$ αို αံုးαα့္αα္αိုα αα္ααွα္းαြα္α αα့္αါαα္။ αံုαါα‘α်α္α‘αα္α‘α $ \displaystyle \text{Theorem 8}$ αဲα $ \displaystyle \text{Theorem 10}$ αို αံုးαိုα α‘αြα့္α‘ေαး αα္းαα္αိုα αα္ααွα္းႏိုα္αါαα္။
α‘ဲαီαိုαြα္αႏိုα္αα္ αα္ေαျααိုα αြα္αူαြားαါαΏαီ။ αα္ေαျααΎαα့္ αေα‘ာα္။
$ \displaystyle \text{Given :}\ \ PT\ \text{is a tangent and }PQR\ \text{is a secant}\text{.}$
$ \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PAQ\text{ is a straight line}\text{.}$
$ \displaystyle \text{To Prove : }P,T,R\ \text{and }B\ \text{are concyclic}\text{.}$
$ \displaystyle \text{Proof : Draw }PB,AB\ \text{and }AR.\text{ }$
$ \displaystyle \text{Since}PT\ \text{is a tangent and }PA\ \text{is a chord of first circle,}$
$ \displaystyle {{\beta }_{1}}=\phi \ (\angle \ \text{between tangent and chord=}\angle \text{ in alt: segment})$
$ \displaystyle \text{And }\theta =\phi +\delta (\ \text{Exterior }\angle \ \text{of }\Delta \text{ = Sum of opposite interior }\angle \text{s)}$
$ \displaystyle \therefore \theta ={{\beta }_{1}}+\delta $
$ \displaystyle \text{Since}ABRQ\ \text{is a cyclic quadrilateral,}$
$ \displaystyle {{\beta }_{2}}+\theta =180{}^\circ $
$ \displaystyle \therefore {{\beta }_{2}}+{{\beta }_{1}}+\delta =180{}^\circ \text{ }$
$ \displaystyle \therefore \angle PBR+\angle PTR=180{}^\circ $
$ \displaystyle \therefore P,T,R\ \text{and }B\ \text{are concyclic}\text{.}$
α
ာαα်αူ၏ α‘αြα်αို αေးα
ားα
ွာα
ောα့်αျှော်αျα်!
Extend ray TP to S. Then,
ReplyDelete\angle SPB = \angle PAB = \angle BRT
and hence, PTRB is cyclic.
α ေα်ာα္းαူေα်ာα္းαားေαြα‘αြα္ αိုαΏαီး straight forward ႃαα ္αα္αα္αα္ ααာ။
nice suggestion!!!
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