How can i Find the Area of a Traingle Formed inside a Triangle?












-1














triangle inside another triangle
Here we have Co Ordinates of A ,B ,C and We have to Find the Area of the Triangle PQR Formed Inside the Triangle ABC.triangle ABC. Point D, E and F divides the sides BC, CA and AB into ratio 1:2 respectively. That is CD=2BD, AE=2CE and BF=2AF. A, D; B, E and C, F are connected. AD and BE intersects at P, BE and CF intersects at Q and CF and AD intersects at R. Please Check the Image in the link.










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  • Angle bisectors are concurrent.
    – Offlaw
    Nov 22 at 13:22










  • Ok but how can I find Point P ,Q R and finally find the area of PQR .. Can you Explain...I don't Understand what You want to say.
    – Raihan Rahman
    Nov 22 at 13:26










  • As angle bisectors intersect at incenter of the triangle then you never get another triangle.
    – Offlaw
    Nov 22 at 13:33












  • Sorry...I think I made a mistake to ask the question...can you please check the image then you can understand what I ask for.
    – Raihan Rahman
    Nov 22 at 13:40










  • Give some more information.
    – Offlaw
    Nov 22 at 13:45
















-1














triangle inside another triangle
Here we have Co Ordinates of A ,B ,C and We have to Find the Area of the Triangle PQR Formed Inside the Triangle ABC.triangle ABC. Point D, E and F divides the sides BC, CA and AB into ratio 1:2 respectively. That is CD=2BD, AE=2CE and BF=2AF. A, D; B, E and C, F are connected. AD and BE intersects at P, BE and CF intersects at Q and CF and AD intersects at R. Please Check the Image in the link.










share|cite|improve this question
























  • Angle bisectors are concurrent.
    – Offlaw
    Nov 22 at 13:22










  • Ok but how can I find Point P ,Q R and finally find the area of PQR .. Can you Explain...I don't Understand what You want to say.
    – Raihan Rahman
    Nov 22 at 13:26










  • As angle bisectors intersect at incenter of the triangle then you never get another triangle.
    – Offlaw
    Nov 22 at 13:33












  • Sorry...I think I made a mistake to ask the question...can you please check the image then you can understand what I ask for.
    – Raihan Rahman
    Nov 22 at 13:40










  • Give some more information.
    – Offlaw
    Nov 22 at 13:45














-1












-1








-1







triangle inside another triangle
Here we have Co Ordinates of A ,B ,C and We have to Find the Area of the Triangle PQR Formed Inside the Triangle ABC.triangle ABC. Point D, E and F divides the sides BC, CA and AB into ratio 1:2 respectively. That is CD=2BD, AE=2CE and BF=2AF. A, D; B, E and C, F are connected. AD and BE intersects at P, BE and CF intersects at Q and CF and AD intersects at R. Please Check the Image in the link.










share|cite|improve this question















triangle inside another triangle
Here we have Co Ordinates of A ,B ,C and We have to Find the Area of the Triangle PQR Formed Inside the Triangle ABC.triangle ABC. Point D, E and F divides the sides BC, CA and AB into ratio 1:2 respectively. That is CD=2BD, AE=2CE and BF=2AF. A, D; B, E and C, F are connected. AD and BE intersects at P, BE and CF intersects at Q and CF and AD intersects at R. Please Check the Image in the link.







geometry triangle coordinate-systems






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edited Nov 22 at 13:58

























asked Nov 22 at 13:16









Raihan Rahman

11




11












  • Angle bisectors are concurrent.
    – Offlaw
    Nov 22 at 13:22










  • Ok but how can I find Point P ,Q R and finally find the area of PQR .. Can you Explain...I don't Understand what You want to say.
    – Raihan Rahman
    Nov 22 at 13:26










  • As angle bisectors intersect at incenter of the triangle then you never get another triangle.
    – Offlaw
    Nov 22 at 13:33












  • Sorry...I think I made a mistake to ask the question...can you please check the image then you can understand what I ask for.
    – Raihan Rahman
    Nov 22 at 13:40










  • Give some more information.
    – Offlaw
    Nov 22 at 13:45


















  • Angle bisectors are concurrent.
    – Offlaw
    Nov 22 at 13:22










  • Ok but how can I find Point P ,Q R and finally find the area of PQR .. Can you Explain...I don't Understand what You want to say.
    – Raihan Rahman
    Nov 22 at 13:26










  • As angle bisectors intersect at incenter of the triangle then you never get another triangle.
    – Offlaw
    Nov 22 at 13:33












  • Sorry...I think I made a mistake to ask the question...can you please check the image then you can understand what I ask for.
    – Raihan Rahman
    Nov 22 at 13:40










  • Give some more information.
    – Offlaw
    Nov 22 at 13:45
















Angle bisectors are concurrent.
– Offlaw
Nov 22 at 13:22




Angle bisectors are concurrent.
– Offlaw
Nov 22 at 13:22












Ok but how can I find Point P ,Q R and finally find the area of PQR .. Can you Explain...I don't Understand what You want to say.
– Raihan Rahman
Nov 22 at 13:26




Ok but how can I find Point P ,Q R and finally find the area of PQR .. Can you Explain...I don't Understand what You want to say.
– Raihan Rahman
Nov 22 at 13:26












As angle bisectors intersect at incenter of the triangle then you never get another triangle.
– Offlaw
Nov 22 at 13:33






As angle bisectors intersect at incenter of the triangle then you never get another triangle.
– Offlaw
Nov 22 at 13:33














Sorry...I think I made a mistake to ask the question...can you please check the image then you can understand what I ask for.
– Raihan Rahman
Nov 22 at 13:40




Sorry...I think I made a mistake to ask the question...can you please check the image then you can understand what I ask for.
– Raihan Rahman
Nov 22 at 13:40












Give some more information.
– Offlaw
Nov 22 at 13:45




Give some more information.
– Offlaw
Nov 22 at 13:45










1 Answer
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Denote by $[ABC]$ the area of the triangle $Delta ABC$.



First note that $$frac{[ADC]}{[ABC]}=frac{overline{DC}}{overline {BC}}=frac{2}{3}Rightarrow [ADC]=frac{2[ABC]}{3}quad (1)$$



Now, by Menelaus's Theorem regarding the triangle $Delta ADB$ and the line through $F,R$ and $C$:
$$frac{overline {AF}}{overline {FB}}*frac{overline {BC}}{overline {DC}}*frac{overline {RD}}{overline {AR}}=1 Rightarrow frac{overline {RD}}{overline {AR}}=frac{overline {FB}}{overline {AF}}*frac{overline {DC}}{overline {BC}}=2*frac{2}{3}=frac{4}{3}$$



Note now that (recall $(1)$) $$frac{overline {RD}}{overline {AR}}=frac{[ADC]-[ARC]}{[ARC]}=frac{[ADC]}{[ARC]}-1=frac{2[ABC]}{3[ARC]}-1=frac{4}{3}$$ $$Rightarrow [ARC]=frac{2[ABC]}{7}$$



Analogously $$[ARC]=[CQB]=[BPA]=frac{2[ABC]}{7}$$
Since $$[PQR]=[ABC]-Bigl([ARC]+[CQB]+[BPA]Bigr)$$
Your have $$[PQR]=[ABC]-3Biggl(frac{2[ABC]}{7}Biggr)=frac{[ABC]}{7}$$



$mathbf {Remark:}$



The problem (configuration) you've suggested is known as the "One-seventh area triangle" and is a special case of "Routh's theorem" which gives a general solution in order to calculate the area of the inner triangle for any three cevians forming this one.






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    1 Answer
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    1 Answer
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    active

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    active

    oldest

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    active

    oldest

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    1














    Denote by $[ABC]$ the area of the triangle $Delta ABC$.



    First note that $$frac{[ADC]}{[ABC]}=frac{overline{DC}}{overline {BC}}=frac{2}{3}Rightarrow [ADC]=frac{2[ABC]}{3}quad (1)$$



    Now, by Menelaus's Theorem regarding the triangle $Delta ADB$ and the line through $F,R$ and $C$:
    $$frac{overline {AF}}{overline {FB}}*frac{overline {BC}}{overline {DC}}*frac{overline {RD}}{overline {AR}}=1 Rightarrow frac{overline {RD}}{overline {AR}}=frac{overline {FB}}{overline {AF}}*frac{overline {DC}}{overline {BC}}=2*frac{2}{3}=frac{4}{3}$$



    Note now that (recall $(1)$) $$frac{overline {RD}}{overline {AR}}=frac{[ADC]-[ARC]}{[ARC]}=frac{[ADC]}{[ARC]}-1=frac{2[ABC]}{3[ARC]}-1=frac{4}{3}$$ $$Rightarrow [ARC]=frac{2[ABC]}{7}$$



    Analogously $$[ARC]=[CQB]=[BPA]=frac{2[ABC]}{7}$$
    Since $$[PQR]=[ABC]-Bigl([ARC]+[CQB]+[BPA]Bigr)$$
    Your have $$[PQR]=[ABC]-3Biggl(frac{2[ABC]}{7}Biggr)=frac{[ABC]}{7}$$



    $mathbf {Remark:}$



    The problem (configuration) you've suggested is known as the "One-seventh area triangle" and is a special case of "Routh's theorem" which gives a general solution in order to calculate the area of the inner triangle for any three cevians forming this one.






    share|cite|improve this answer


























      1














      Denote by $[ABC]$ the area of the triangle $Delta ABC$.



      First note that $$frac{[ADC]}{[ABC]}=frac{overline{DC}}{overline {BC}}=frac{2}{3}Rightarrow [ADC]=frac{2[ABC]}{3}quad (1)$$



      Now, by Menelaus's Theorem regarding the triangle $Delta ADB$ and the line through $F,R$ and $C$:
      $$frac{overline {AF}}{overline {FB}}*frac{overline {BC}}{overline {DC}}*frac{overline {RD}}{overline {AR}}=1 Rightarrow frac{overline {RD}}{overline {AR}}=frac{overline {FB}}{overline {AF}}*frac{overline {DC}}{overline {BC}}=2*frac{2}{3}=frac{4}{3}$$



      Note now that (recall $(1)$) $$frac{overline {RD}}{overline {AR}}=frac{[ADC]-[ARC]}{[ARC]}=frac{[ADC]}{[ARC]}-1=frac{2[ABC]}{3[ARC]}-1=frac{4}{3}$$ $$Rightarrow [ARC]=frac{2[ABC]}{7}$$



      Analogously $$[ARC]=[CQB]=[BPA]=frac{2[ABC]}{7}$$
      Since $$[PQR]=[ABC]-Bigl([ARC]+[CQB]+[BPA]Bigr)$$
      Your have $$[PQR]=[ABC]-3Biggl(frac{2[ABC]}{7}Biggr)=frac{[ABC]}{7}$$



      $mathbf {Remark:}$



      The problem (configuration) you've suggested is known as the "One-seventh area triangle" and is a special case of "Routh's theorem" which gives a general solution in order to calculate the area of the inner triangle for any three cevians forming this one.






      share|cite|improve this answer
























        1












        1








        1






        Denote by $[ABC]$ the area of the triangle $Delta ABC$.



        First note that $$frac{[ADC]}{[ABC]}=frac{overline{DC}}{overline {BC}}=frac{2}{3}Rightarrow [ADC]=frac{2[ABC]}{3}quad (1)$$



        Now, by Menelaus's Theorem regarding the triangle $Delta ADB$ and the line through $F,R$ and $C$:
        $$frac{overline {AF}}{overline {FB}}*frac{overline {BC}}{overline {DC}}*frac{overline {RD}}{overline {AR}}=1 Rightarrow frac{overline {RD}}{overline {AR}}=frac{overline {FB}}{overline {AF}}*frac{overline {DC}}{overline {BC}}=2*frac{2}{3}=frac{4}{3}$$



        Note now that (recall $(1)$) $$frac{overline {RD}}{overline {AR}}=frac{[ADC]-[ARC]}{[ARC]}=frac{[ADC]}{[ARC]}-1=frac{2[ABC]}{3[ARC]}-1=frac{4}{3}$$ $$Rightarrow [ARC]=frac{2[ABC]}{7}$$



        Analogously $$[ARC]=[CQB]=[BPA]=frac{2[ABC]}{7}$$
        Since $$[PQR]=[ABC]-Bigl([ARC]+[CQB]+[BPA]Bigr)$$
        Your have $$[PQR]=[ABC]-3Biggl(frac{2[ABC]}{7}Biggr)=frac{[ABC]}{7}$$



        $mathbf {Remark:}$



        The problem (configuration) you've suggested is known as the "One-seventh area triangle" and is a special case of "Routh's theorem" which gives a general solution in order to calculate the area of the inner triangle for any three cevians forming this one.






        share|cite|improve this answer












        Denote by $[ABC]$ the area of the triangle $Delta ABC$.



        First note that $$frac{[ADC]}{[ABC]}=frac{overline{DC}}{overline {BC}}=frac{2}{3}Rightarrow [ADC]=frac{2[ABC]}{3}quad (1)$$



        Now, by Menelaus's Theorem regarding the triangle $Delta ADB$ and the line through $F,R$ and $C$:
        $$frac{overline {AF}}{overline {FB}}*frac{overline {BC}}{overline {DC}}*frac{overline {RD}}{overline {AR}}=1 Rightarrow frac{overline {RD}}{overline {AR}}=frac{overline {FB}}{overline {AF}}*frac{overline {DC}}{overline {BC}}=2*frac{2}{3}=frac{4}{3}$$



        Note now that (recall $(1)$) $$frac{overline {RD}}{overline {AR}}=frac{[ADC]-[ARC]}{[ARC]}=frac{[ADC]}{[ARC]}-1=frac{2[ABC]}{3[ARC]}-1=frac{4}{3}$$ $$Rightarrow [ARC]=frac{2[ABC]}{7}$$



        Analogously $$[ARC]=[CQB]=[BPA]=frac{2[ABC]}{7}$$
        Since $$[PQR]=[ABC]-Bigl([ARC]+[CQB]+[BPA]Bigr)$$
        Your have $$[PQR]=[ABC]-3Biggl(frac{2[ABC]}{7}Biggr)=frac{[ABC]}{7}$$



        $mathbf {Remark:}$



        The problem (configuration) you've suggested is known as the "One-seventh area triangle" and is a special case of "Routh's theorem" which gives a general solution in order to calculate the area of the inner triangle for any three cevians forming this one.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 22 at 20:39









        Dr. Mathva

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