document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1033395>Question 1033395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a parallelogram ABCD and and a line l outside the parallelogram. Let the distance from points A,B,C,D to line l be a,b,c,D respectively. Prove that a+c = b+d </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #648005 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=648005\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since in the Euclidean system, (absolute) distance between objects is invariant under translation/rotation/reflection of axes, we can assume that the line L in the problem is the x-axis and the parallelogram lies above the x-axis.  (If L is parallel to the x-axis, just translate the latter upward or downward; if L intersects the x-axis at point P, rotate the x-axis around the point P to correspond to line L; if the parallelogram is below the line L, reflect the former so that it is above the line L.)\r<BR>\n" );
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document.write( "Now let the leftmost vertex be A(a,b).  Then point B (assuming clockwise direction) will be (a + r, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bm%5B1%5D%2Ar ALIGN=MIDDLE  ALT=\"b%2Bm%5B1%5D%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >), where r is any positive real number and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D ALIGN=MIDDLE  ALT=\"m%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope of the line from point A to B.\r<BR>\n" );
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document.write( "Point D will have coordinates (a+s, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bm%5B2%5D%2As ALIGN=MIDDLE  ALT=\"b%2Bm%5B2%5D%2As\"   DISABLED_style= \"max-width:90%; height:auto;\" >), where s again any arbitrary positive real number, while <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B2%5D ALIGN=MIDDLE  ALT=\"m%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope of the line from point A to point D.\r<BR>\n" );
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document.write( "Finally point C will have coordinates (a+r+s, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bm%5B1%5D%2Ar+%2Bm%5B2%5D%2As ALIGN=MIDDLE  ALT=\"b%2Bm%5B1%5D%2Ar+%2Bm%5B2%5D%2As\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  The line  AB, which has slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D ALIGN=MIDDLE  ALT=\"m%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, will then have the same slope as line DC.\r<BR>\n" );
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document.write( "Now the distance of point A from the x-axis is b.<BR>\n" );
document.write( "The distance of point B from the x-axis is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bm%5B1%5D%2Ar ALIGN=MIDDLE  ALT=\"b%2Bm%5B1%5D%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The distance of point C from the x-axis is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bm%5B1%5D%2Ar+%2Bm%5B2%5D%2As ALIGN=MIDDLE  ALT=\"b%2Bm%5B1%5D%2Ar+%2Bm%5B2%5D%2As\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The distance of point D from the x-axis is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bm%5B2%5D%2As ALIGN=MIDDLE  ALT=\"b%2Bm%5B2%5D%2As\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "==> a+c = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%2Bb%2Bm%5B1%5D%2Ar+%2Bm%5B2%5D%2As+=+2b%2Bm%5B1%5D%2Ar+%2Bm%5B2%5D%2As+ ALIGN=MIDDLE  ALT=\"b+%2Bb%2Bm%5B1%5D%2Ar+%2Bm%5B2%5D%2As+=+2b%2Bm%5B1%5D%2Ar+%2Bm%5B2%5D%2As+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "and b + d = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bm%5B1%5D%2Ar+%2B+b%2Bm%5B2%5D%2As+=+2b+%2B+m%5B1%5D%2Ar+%2Bm%5B2%5D%2As ALIGN=MIDDLE  ALT=\"b%2Bm%5B1%5D%2Ar+%2B+b%2Bm%5B2%5D%2As+=+2b+%2B+m%5B1%5D%2Ar+%2Bm%5B2%5D%2As\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "AND the result immediately follows.<BR>\n" );
document.write( " </SPAN>\n" );
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