document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1075375>Question 1075375</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the flaw in the following argument purporting to construct a rectangle. Let A and B be any two points. There is a line l through A perpendicular to line AB (proposition 3.16) and, similarly, there is a line m through B perpendicular to line AB. Take any point C on m other than B. There is a line through C perpendicular to l- let it intersect l at D. Then ABCD is a rectangle.\r<BR>\n" );
document.write( "\n" );
document.write( "Proposition 3.16: For every line l and every point P there exists a line P perpendicular to l. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #690092 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=690092\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let AB be a line along the length of the (horizontal) base of a shoe box.<BR>\n" );
document.write( "Let l be a line through A along the width of the base of the shoe box.<BR>\n" );
document.write( "You are going to construct line m perpendicular to AB.<BR>\n" );
document.write( "There is no requirement for that line to be parallel to l. <BR>\n" );
document.write( "It does not need to be on the same plane determined by AB and l.<BR>\n" );
document.write( "Let m be a vertical line through B.<BR>\n" );
document.write( "lines l and m are both perpendicular to line AB.<BR>\n" );
document.write( "They do not intersect, but they are not parallel<BR>\n" );
document.write( "(or even in the same plane).<BR>\n" );
document.write( "Take any point C on m, other than B.<BR>\n" );
document.write( "Points A, B and C determine a plane perpendicular to line l.<BR>\n" );
document.write( "All lines on that plane that go through A are perpendicular to line l.<BR>\n" );
document.write( "Line AC is one of them. It intersects line l at point D, which happens to be point A.<BR>\n" );
document.write( "You have just constructed triangle ABC.<BR>\n" );
document.write( "You have other options for a line m that is not coplanar with l,<BR>\n" );
document.write( "and may give you a point D different from A,<BR>\n" );
document.write( "but ABCD will not be planar, so it will not be a rectangle. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );