Prove that $angle DAP=angle CAB$ in a parallelogram $ABCD$












4












$begingroup$


Let $ABCD$ be a parallelogram, and let $K$ be on $BC$ and $L$ on $CD$ so that $BKcdot BC=DLcdot DC$. Let point $P$ be where $DK$ and $BL$ intersect. Prove that $angle DAP=angle CAB$ (angles $DAP$ and $CAB$ are equal).



I got that $ADL$ and $ABC$ are similar so it is enough to prove that $angle LAP=angle CAK$, but I think there is an better way to use the given statement.










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$endgroup$












  • $begingroup$
    docs.google.com/file/d/0ByGmeK3txyClSUM5OFNoSmQ5R1E/…. Here is a picture I drew, bcs I dont have a pc nearby.
    $endgroup$
    – HeatTheIce
    Jan 23 '15 at 15:03












  • $begingroup$
    see if this works. extend $AP$ to meet $BC$ at $Q.$ seems that $AKQ$ is similar to $ALC$
    $endgroup$
    – abel
    Jan 23 '15 at 16:53










  • $begingroup$
    i have verified that the result is certainly true for a rectangle $ABCD$
    $endgroup$
    – abel
    Jan 23 '15 at 17:19










  • $begingroup$
    For those looking to construct this figure: Take $O$ a point on the perpendicular bisector of $overline{BD}$, and consider $bigcirc O$ that passes through $C$. Then $K$ and $L$ are the other points where $bigcirc O$ passes through the specified edges of the parallelogram.
    $endgroup$
    – Blue
    Jan 23 '15 at 18:07












  • $begingroup$
    Well, we can just draw a ray from A forming an angle with AC equal to CAB, pick any point P on that ray, and join P with C and with B to get K and L. By the way, @Blue can you check your construction? I think it looks very suspicious. If we put the parallel translations K' and L' of K and L on the sides AD and AB, then I see a circle through BDK'L'. Those circles on C that Jack and you constructed, I don't see them.
    $endgroup$
    – Pp..
    Jan 23 '15 at 18:46
















4












$begingroup$


Let $ABCD$ be a parallelogram, and let $K$ be on $BC$ and $L$ on $CD$ so that $BKcdot BC=DLcdot DC$. Let point $P$ be where $DK$ and $BL$ intersect. Prove that $angle DAP=angle CAB$ (angles $DAP$ and $CAB$ are equal).



I got that $ADL$ and $ABC$ are similar so it is enough to prove that $angle LAP=angle CAK$, but I think there is an better way to use the given statement.










share|cite|improve this question











$endgroup$












  • $begingroup$
    docs.google.com/file/d/0ByGmeK3txyClSUM5OFNoSmQ5R1E/…. Here is a picture I drew, bcs I dont have a pc nearby.
    $endgroup$
    – HeatTheIce
    Jan 23 '15 at 15:03












  • $begingroup$
    see if this works. extend $AP$ to meet $BC$ at $Q.$ seems that $AKQ$ is similar to $ALC$
    $endgroup$
    – abel
    Jan 23 '15 at 16:53










  • $begingroup$
    i have verified that the result is certainly true for a rectangle $ABCD$
    $endgroup$
    – abel
    Jan 23 '15 at 17:19










  • $begingroup$
    For those looking to construct this figure: Take $O$ a point on the perpendicular bisector of $overline{BD}$, and consider $bigcirc O$ that passes through $C$. Then $K$ and $L$ are the other points where $bigcirc O$ passes through the specified edges of the parallelogram.
    $endgroup$
    – Blue
    Jan 23 '15 at 18:07












  • $begingroup$
    Well, we can just draw a ray from A forming an angle with AC equal to CAB, pick any point P on that ray, and join P with C and with B to get K and L. By the way, @Blue can you check your construction? I think it looks very suspicious. If we put the parallel translations K' and L' of K and L on the sides AD and AB, then I see a circle through BDK'L'. Those circles on C that Jack and you constructed, I don't see them.
    $endgroup$
    – Pp..
    Jan 23 '15 at 18:46














4












4








4


4



$begingroup$


Let $ABCD$ be a parallelogram, and let $K$ be on $BC$ and $L$ on $CD$ so that $BKcdot BC=DLcdot DC$. Let point $P$ be where $DK$ and $BL$ intersect. Prove that $angle DAP=angle CAB$ (angles $DAP$ and $CAB$ are equal).



I got that $ADL$ and $ABC$ are similar so it is enough to prove that $angle LAP=angle CAK$, but I think there is an better way to use the given statement.










share|cite|improve this question











$endgroup$




Let $ABCD$ be a parallelogram, and let $K$ be on $BC$ and $L$ on $CD$ so that $BKcdot BC=DLcdot DC$. Let point $P$ be where $DK$ and $BL$ intersect. Prove that $angle DAP=angle CAB$ (angles $DAP$ and $CAB$ are equal).



I got that $ADL$ and $ABC$ are similar so it is enough to prove that $angle LAP=angle CAK$, but I think there is an better way to use the given statement.







geometry euclidean-geometry projective-geometry geometric-transformation






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share|cite|improve this question













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share|cite|improve this question








edited Dec 29 '18 at 15:11









Maria Mazur

46.9k1260120




46.9k1260120










asked Jan 23 '15 at 14:23









HeatTheIceHeatTheIce

1,775618




1,775618












  • $begingroup$
    docs.google.com/file/d/0ByGmeK3txyClSUM5OFNoSmQ5R1E/…. Here is a picture I drew, bcs I dont have a pc nearby.
    $endgroup$
    – HeatTheIce
    Jan 23 '15 at 15:03












  • $begingroup$
    see if this works. extend $AP$ to meet $BC$ at $Q.$ seems that $AKQ$ is similar to $ALC$
    $endgroup$
    – abel
    Jan 23 '15 at 16:53










  • $begingroup$
    i have verified that the result is certainly true for a rectangle $ABCD$
    $endgroup$
    – abel
    Jan 23 '15 at 17:19










  • $begingroup$
    For those looking to construct this figure: Take $O$ a point on the perpendicular bisector of $overline{BD}$, and consider $bigcirc O$ that passes through $C$. Then $K$ and $L$ are the other points where $bigcirc O$ passes through the specified edges of the parallelogram.
    $endgroup$
    – Blue
    Jan 23 '15 at 18:07












  • $begingroup$
    Well, we can just draw a ray from A forming an angle with AC equal to CAB, pick any point P on that ray, and join P with C and with B to get K and L. By the way, @Blue can you check your construction? I think it looks very suspicious. If we put the parallel translations K' and L' of K and L on the sides AD and AB, then I see a circle through BDK'L'. Those circles on C that Jack and you constructed, I don't see them.
    $endgroup$
    – Pp..
    Jan 23 '15 at 18:46


















  • $begingroup$
    docs.google.com/file/d/0ByGmeK3txyClSUM5OFNoSmQ5R1E/…. Here is a picture I drew, bcs I dont have a pc nearby.
    $endgroup$
    – HeatTheIce
    Jan 23 '15 at 15:03












  • $begingroup$
    see if this works. extend $AP$ to meet $BC$ at $Q.$ seems that $AKQ$ is similar to $ALC$
    $endgroup$
    – abel
    Jan 23 '15 at 16:53










  • $begingroup$
    i have verified that the result is certainly true for a rectangle $ABCD$
    $endgroup$
    – abel
    Jan 23 '15 at 17:19










  • $begingroup$
    For those looking to construct this figure: Take $O$ a point on the perpendicular bisector of $overline{BD}$, and consider $bigcirc O$ that passes through $C$. Then $K$ and $L$ are the other points where $bigcirc O$ passes through the specified edges of the parallelogram.
    $endgroup$
    – Blue
    Jan 23 '15 at 18:07












  • $begingroup$
    Well, we can just draw a ray from A forming an angle with AC equal to CAB, pick any point P on that ray, and join P with C and with B to get K and L. By the way, @Blue can you check your construction? I think it looks very suspicious. If we put the parallel translations K' and L' of K and L on the sides AD and AB, then I see a circle through BDK'L'. Those circles on C that Jack and you constructed, I don't see them.
    $endgroup$
    – Pp..
    Jan 23 '15 at 18:46
















$begingroup$
docs.google.com/file/d/0ByGmeK3txyClSUM5OFNoSmQ5R1E/…. Here is a picture I drew, bcs I dont have a pc nearby.
$endgroup$
– HeatTheIce
Jan 23 '15 at 15:03






$begingroup$
docs.google.com/file/d/0ByGmeK3txyClSUM5OFNoSmQ5R1E/…. Here is a picture I drew, bcs I dont have a pc nearby.
$endgroup$
– HeatTheIce
Jan 23 '15 at 15:03














$begingroup$
see if this works. extend $AP$ to meet $BC$ at $Q.$ seems that $AKQ$ is similar to $ALC$
$endgroup$
– abel
Jan 23 '15 at 16:53




$begingroup$
see if this works. extend $AP$ to meet $BC$ at $Q.$ seems that $AKQ$ is similar to $ALC$
$endgroup$
– abel
Jan 23 '15 at 16:53












$begingroup$
i have verified that the result is certainly true for a rectangle $ABCD$
$endgroup$
– abel
Jan 23 '15 at 17:19




$begingroup$
i have verified that the result is certainly true for a rectangle $ABCD$
$endgroup$
– abel
Jan 23 '15 at 17:19












$begingroup$
For those looking to construct this figure: Take $O$ a point on the perpendicular bisector of $overline{BD}$, and consider $bigcirc O$ that passes through $C$. Then $K$ and $L$ are the other points where $bigcirc O$ passes through the specified edges of the parallelogram.
$endgroup$
– Blue
Jan 23 '15 at 18:07






$begingroup$
For those looking to construct this figure: Take $O$ a point on the perpendicular bisector of $overline{BD}$, and consider $bigcirc O$ that passes through $C$. Then $K$ and $L$ are the other points where $bigcirc O$ passes through the specified edges of the parallelogram.
$endgroup$
– Blue
Jan 23 '15 at 18:07














$begingroup$
Well, we can just draw a ray from A forming an angle with AC equal to CAB, pick any point P on that ray, and join P with C and with B to get K and L. By the way, @Blue can you check your construction? I think it looks very suspicious. If we put the parallel translations K' and L' of K and L on the sides AD and AB, then I see a circle through BDK'L'. Those circles on C that Jack and you constructed, I don't see them.
$endgroup$
– Pp..
Jan 23 '15 at 18:46




$begingroup$
Well, we can just draw a ray from A forming an angle with AC equal to CAB, pick any point P on that ray, and join P with C and with B to get K and L. By the way, @Blue can you check your construction? I think it looks very suspicious. If we put the parallel translations K' and L' of K and L on the sides AD and AB, then I see a circle through BDK'L'. Those circles on C that Jack and you constructed, I don't see them.
$endgroup$
– Pp..
Jan 23 '15 at 18:46










2 Answers
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1












$begingroup$

This is a solution using oblique coordinates.




The main purpose is to show that the proof is entirely linear. This means that one can construct a synthetic geometry proof using only similarity of triangles, after drawing lines through $K,L,P$ parallel to the sides of the parallelogram.




Let's use $AB$ and $AD$ as the axes. So $A=(0,0)$, $B=(0,ell_1)$, $D=(ell_2,0)$ and $C=(ell_2,ell_1)$. Call $r:=frac{ell_1}{ell_2}$.



The line $AC$ is $y=rx$.




We want to prove that the line $y=frac{1}{r}x$ is the locus of the points $P$. This is enough to get the conclusion.




Take a point $P:=(a,a/r)$ on this line.



Then the line from $D=(ell_2,0)$ to $P=(a,a/r)$ is given by
$$y=frac{frac{a}{r}-0}{a-ell_2}x-ell_2frac{frac{a}{r}-0}{a-ell_2}$$



This intersects $BC=(y=ell_1)$ at $K=left(left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a},ell_1right)$, i.e. $$BK=left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}$$



The line from $B=(0,ell_1)$ to $P=(a,a/r)$ is given by
$$y=frac{frac{a}{r}-ell_1}{a-0}x+ell_1$$
This line intersects $DC=(x=ell_2)$ at $L=left(ell_2,frac{a-rell_1}{ra}ell_2+ell_1right)$, i.e.
$$DL=frac{a-rell_1}{ra}ell_2+ell_1$$



Now we only need to divide



$$frac{BK}{DL}=frac{left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}}{frac{a-rell_1}{ra}ell_2+ell_1}=r=frac{ell_1}{ell_2}=frac{DC}{BC}$$






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$endgroup$





















    1












    $begingroup$

    With knowledge of projective geometry this is pretty easy problem.
    enter image description here
    Let $a=AB$, $b = BC$, $BK = x$ and $DL = y$. Then we have $$y= {bover a} x$$
    so $L$ is lineary dependent on $K$, thus the transformation $$Kmapsto L$$ from a line $BC$ to a line $CD$ is projective. This one induces new projective transformation $$DK mapsto BL$$ from a bundle of lines through $D$ to a bundle of lines through $B$. Since clearly $BD$ goes to it self, this transformation is perspective so the intersection point $P$ describes some line.



    Clearly when $K=infty $ then $P=A$ and let $C$ maps to $E$ (so $DE = {b^2over a}$). We see that $E$ must be on $CD$. All we have to see now is that the triangle $ADE$ is similar to the triangle $ABC$ which is simple since: $${DEover DA} = {{b^2over a }over b} = {bover a} = {BCover BA}$$






    share|cite|improve this answer











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      2 Answers
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      2 Answers
      2






      active

      oldest

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      active

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      active

      oldest

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      1












      $begingroup$

      This is a solution using oblique coordinates.




      The main purpose is to show that the proof is entirely linear. This means that one can construct a synthetic geometry proof using only similarity of triangles, after drawing lines through $K,L,P$ parallel to the sides of the parallelogram.




      Let's use $AB$ and $AD$ as the axes. So $A=(0,0)$, $B=(0,ell_1)$, $D=(ell_2,0)$ and $C=(ell_2,ell_1)$. Call $r:=frac{ell_1}{ell_2}$.



      The line $AC$ is $y=rx$.




      We want to prove that the line $y=frac{1}{r}x$ is the locus of the points $P$. This is enough to get the conclusion.




      Take a point $P:=(a,a/r)$ on this line.



      Then the line from $D=(ell_2,0)$ to $P=(a,a/r)$ is given by
      $$y=frac{frac{a}{r}-0}{a-ell_2}x-ell_2frac{frac{a}{r}-0}{a-ell_2}$$



      This intersects $BC=(y=ell_1)$ at $K=left(left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a},ell_1right)$, i.e. $$BK=left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}$$



      The line from $B=(0,ell_1)$ to $P=(a,a/r)$ is given by
      $$y=frac{frac{a}{r}-ell_1}{a-0}x+ell_1$$
      This line intersects $DC=(x=ell_2)$ at $L=left(ell_2,frac{a-rell_1}{ra}ell_2+ell_1right)$, i.e.
      $$DL=frac{a-rell_1}{ra}ell_2+ell_1$$



      Now we only need to divide



      $$frac{BK}{DL}=frac{left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}}{frac{a-rell_1}{ra}ell_2+ell_1}=r=frac{ell_1}{ell_2}=frac{DC}{BC}$$






      share|cite|improve this answer











      $endgroup$


















        1












        $begingroup$

        This is a solution using oblique coordinates.




        The main purpose is to show that the proof is entirely linear. This means that one can construct a synthetic geometry proof using only similarity of triangles, after drawing lines through $K,L,P$ parallel to the sides of the parallelogram.




        Let's use $AB$ and $AD$ as the axes. So $A=(0,0)$, $B=(0,ell_1)$, $D=(ell_2,0)$ and $C=(ell_2,ell_1)$. Call $r:=frac{ell_1}{ell_2}$.



        The line $AC$ is $y=rx$.




        We want to prove that the line $y=frac{1}{r}x$ is the locus of the points $P$. This is enough to get the conclusion.




        Take a point $P:=(a,a/r)$ on this line.



        Then the line from $D=(ell_2,0)$ to $P=(a,a/r)$ is given by
        $$y=frac{frac{a}{r}-0}{a-ell_2}x-ell_2frac{frac{a}{r}-0}{a-ell_2}$$



        This intersects $BC=(y=ell_1)$ at $K=left(left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a},ell_1right)$, i.e. $$BK=left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}$$



        The line from $B=(0,ell_1)$ to $P=(a,a/r)$ is given by
        $$y=frac{frac{a}{r}-ell_1}{a-0}x+ell_1$$
        This line intersects $DC=(x=ell_2)$ at $L=left(ell_2,frac{a-rell_1}{ra}ell_2+ell_1right)$, i.e.
        $$DL=frac{a-rell_1}{ra}ell_2+ell_1$$



        Now we only need to divide



        $$frac{BK}{DL}=frac{left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}}{frac{a-rell_1}{ra}ell_2+ell_1}=r=frac{ell_1}{ell_2}=frac{DC}{BC}$$






        share|cite|improve this answer











        $endgroup$
















          1












          1








          1





          $begingroup$

          This is a solution using oblique coordinates.




          The main purpose is to show that the proof is entirely linear. This means that one can construct a synthetic geometry proof using only similarity of triangles, after drawing lines through $K,L,P$ parallel to the sides of the parallelogram.




          Let's use $AB$ and $AD$ as the axes. So $A=(0,0)$, $B=(0,ell_1)$, $D=(ell_2,0)$ and $C=(ell_2,ell_1)$. Call $r:=frac{ell_1}{ell_2}$.



          The line $AC$ is $y=rx$.




          We want to prove that the line $y=frac{1}{r}x$ is the locus of the points $P$. This is enough to get the conclusion.




          Take a point $P:=(a,a/r)$ on this line.



          Then the line from $D=(ell_2,0)$ to $P=(a,a/r)$ is given by
          $$y=frac{frac{a}{r}-0}{a-ell_2}x-ell_2frac{frac{a}{r}-0}{a-ell_2}$$



          This intersects $BC=(y=ell_1)$ at $K=left(left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a},ell_1right)$, i.e. $$BK=left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}$$



          The line from $B=(0,ell_1)$ to $P=(a,a/r)$ is given by
          $$y=frac{frac{a}{r}-ell_1}{a-0}x+ell_1$$
          This line intersects $DC=(x=ell_2)$ at $L=left(ell_2,frac{a-rell_1}{ra}ell_2+ell_1right)$, i.e.
          $$DL=frac{a-rell_1}{ra}ell_2+ell_1$$



          Now we only need to divide



          $$frac{BK}{DL}=frac{left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}}{frac{a-rell_1}{ra}ell_2+ell_1}=r=frac{ell_1}{ell_2}=frac{DC}{BC}$$






          share|cite|improve this answer











          $endgroup$



          This is a solution using oblique coordinates.




          The main purpose is to show that the proof is entirely linear. This means that one can construct a synthetic geometry proof using only similarity of triangles, after drawing lines through $K,L,P$ parallel to the sides of the parallelogram.




          Let's use $AB$ and $AD$ as the axes. So $A=(0,0)$, $B=(0,ell_1)$, $D=(ell_2,0)$ and $C=(ell_2,ell_1)$. Call $r:=frac{ell_1}{ell_2}$.



          The line $AC$ is $y=rx$.




          We want to prove that the line $y=frac{1}{r}x$ is the locus of the points $P$. This is enough to get the conclusion.




          Take a point $P:=(a,a/r)$ on this line.



          Then the line from $D=(ell_2,0)$ to $P=(a,a/r)$ is given by
          $$y=frac{frac{a}{r}-0}{a-ell_2}x-ell_2frac{frac{a}{r}-0}{a-ell_2}$$



          This intersects $BC=(y=ell_1)$ at $K=left(left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a},ell_1right)$, i.e. $$BK=left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}$$



          The line from $B=(0,ell_1)$ to $P=(a,a/r)$ is given by
          $$y=frac{frac{a}{r}-ell_1}{a-0}x+ell_1$$
          This line intersects $DC=(x=ell_2)$ at $L=left(ell_2,frac{a-rell_1}{ra}ell_2+ell_1right)$, i.e.
          $$DL=frac{a-rell_1}{ra}ell_2+ell_1$$



          Now we only need to divide



          $$frac{BK}{DL}=frac{left[ell_1+frac{ell_2a}{r(a-ell_2)}right]cdotfrac{r(a-ell_2)}{a}}{frac{a-rell_1}{ra}ell_2+ell_1}=r=frac{ell_1}{ell_2}=frac{DC}{BC}$$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 25 '15 at 5:39

























          answered Jan 24 '15 at 5:02









          Pp..Pp..

          5,1791927




          5,1791927























              1












              $begingroup$

              With knowledge of projective geometry this is pretty easy problem.
              enter image description here
              Let $a=AB$, $b = BC$, $BK = x$ and $DL = y$. Then we have $$y= {bover a} x$$
              so $L$ is lineary dependent on $K$, thus the transformation $$Kmapsto L$$ from a line $BC$ to a line $CD$ is projective. This one induces new projective transformation $$DK mapsto BL$$ from a bundle of lines through $D$ to a bundle of lines through $B$. Since clearly $BD$ goes to it self, this transformation is perspective so the intersection point $P$ describes some line.



              Clearly when $K=infty $ then $P=A$ and let $C$ maps to $E$ (so $DE = {b^2over a}$). We see that $E$ must be on $CD$. All we have to see now is that the triangle $ADE$ is similar to the triangle $ABC$ which is simple since: $${DEover DA} = {{b^2over a }over b} = {bover a} = {BCover BA}$$






              share|cite|improve this answer











              $endgroup$


















                1












                $begingroup$

                With knowledge of projective geometry this is pretty easy problem.
                enter image description here
                Let $a=AB$, $b = BC$, $BK = x$ and $DL = y$. Then we have $$y= {bover a} x$$
                so $L$ is lineary dependent on $K$, thus the transformation $$Kmapsto L$$ from a line $BC$ to a line $CD$ is projective. This one induces new projective transformation $$DK mapsto BL$$ from a bundle of lines through $D$ to a bundle of lines through $B$. Since clearly $BD$ goes to it self, this transformation is perspective so the intersection point $P$ describes some line.



                Clearly when $K=infty $ then $P=A$ and let $C$ maps to $E$ (so $DE = {b^2over a}$). We see that $E$ must be on $CD$. All we have to see now is that the triangle $ADE$ is similar to the triangle $ABC$ which is simple since: $${DEover DA} = {{b^2over a }over b} = {bover a} = {BCover BA}$$






                share|cite|improve this answer











                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  With knowledge of projective geometry this is pretty easy problem.
                  enter image description here
                  Let $a=AB$, $b = BC$, $BK = x$ and $DL = y$. Then we have $$y= {bover a} x$$
                  so $L$ is lineary dependent on $K$, thus the transformation $$Kmapsto L$$ from a line $BC$ to a line $CD$ is projective. This one induces new projective transformation $$DK mapsto BL$$ from a bundle of lines through $D$ to a bundle of lines through $B$. Since clearly $BD$ goes to it self, this transformation is perspective so the intersection point $P$ describes some line.



                  Clearly when $K=infty $ then $P=A$ and let $C$ maps to $E$ (so $DE = {b^2over a}$). We see that $E$ must be on $CD$. All we have to see now is that the triangle $ADE$ is similar to the triangle $ABC$ which is simple since: $${DEover DA} = {{b^2over a }over b} = {bover a} = {BCover BA}$$






                  share|cite|improve this answer











                  $endgroup$



                  With knowledge of projective geometry this is pretty easy problem.
                  enter image description here
                  Let $a=AB$, $b = BC$, $BK = x$ and $DL = y$. Then we have $$y= {bover a} x$$
                  so $L$ is lineary dependent on $K$, thus the transformation $$Kmapsto L$$ from a line $BC$ to a line $CD$ is projective. This one induces new projective transformation $$DK mapsto BL$$ from a bundle of lines through $D$ to a bundle of lines through $B$. Since clearly $BD$ goes to it self, this transformation is perspective so the intersection point $P$ describes some line.



                  Clearly when $K=infty $ then $P=A$ and let $C$ maps to $E$ (so $DE = {b^2over a}$). We see that $E$ must be on $CD$. All we have to see now is that the triangle $ADE$ is similar to the triangle $ABC$ which is simple since: $${DEover DA} = {{b^2over a }over b} = {bover a} = {BCover BA}$$







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 25 '18 at 18:46

























                  answered Dec 25 '18 at 18:40









                  Maria MazurMaria Mazur

                  46.9k1260120




                  46.9k1260120






























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