document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192322>Question 1192322</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Suppose you have n points, no three of which are collinear. How many lines contain two of these n points?<BR>\n" );
document.write( "2. If no four of the n points are coplanar, how many planes contain three of the n points?<BR>\n" );
document.write( "Hint: (for 3 and 4, generalize in a form of a formula)<BR>\n" );
document.write( "3.Prove theorem 1.1.4. The steps in the proof are already given: you just have to supply the reasons for each step.<BR>\n" );
document.write( "Theorem 1.1.4. If two lines intersect, then their union lies in exactly one plane.<BR>\n" );
document.write( "Proof: Let and be two intersecting lines.<BR>\n" );
document.write( "a. A ∩ B is a point p.<BR>\n" );
document.write( "b. B contains a point q ≠ p.<BR>\n" );
document.write( "c. There is a plane E, containing A and q.<BR>\n" );
document.write( "d. E contains A ∪ B.<BR>\n" );
document.write( "e. No other plane contains A ∪ B. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848966 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848966\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's break down these geometry problems:\r<BR>\n" );
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document.write( "**1. Number of Lines from n Points**\r<BR>\n" );
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document.write( "* **Concept:** To form a line, we need to select 2 points from the given 'n' points. <BR>\n" );
document.write( "* **Calculation:** The number of ways to select 2 points from 'n' points is given by the combination formula: <BR>\n" );
document.write( "    * nC2 = n! / (2! * (n-2)!) <BR>\n" );
document.write( "    * where 'n!' denotes the factorial of n (n! = n * (n-1) * (n-2) * ... * 1)\r<BR>\n" );
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document.write( "**Therefore, the number of lines that can be formed from 'n' non-collinear points is nC2.**\r<BR>\n" );
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document.write( "**2. Number of Planes from n Points**\r<BR>\n" );
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document.write( "* **Concept:** To form a plane, we need to select 3 non-collinear points.<BR>\n" );
document.write( "* **Calculation:** The number of ways to select 3 points from 'n' points is given by: <BR>\n" );
document.write( "    * nC3 = n! / (3! * (n-3)!)\r<BR>\n" );
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document.write( "**Therefore, the number of planes that can be formed from 'n' non-coplanar points is nC3.**\r<BR>\n" );
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document.write( "**3. Proof of Theorem 1.1.4**\r<BR>\n" );
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document.write( "* **a. A ∩ B is a point p.** <BR>\n" );
document.write( "    * **Reason:** By definition, intersecting lines share exactly one point in common.\r<BR>\n" );
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document.write( "* **b. B contains a point q ≠ p.**<BR>\n" );
document.write( "    * **Reason:** Since B is a line, it must contain at least two distinct points. \r<BR>\n" );
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document.write( "* **c. There is a plane E, containing A and q.**<BR>\n" );
document.write( "    * **Reason:** Two distinct points (p and q) determine a unique line. A line and a point not on that line uniquely determine a plane.\r<BR>\n" );
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document.write( "* **d. E contains A ∪ B.**<BR>\n" );
document.write( "    * **Reason:** <BR>\n" );
document.write( "        * E contains line A (by definition).<BR>\n" );
document.write( "        * E contains point q (by definition).<BR>\n" );
document.write( "        * Since E contains a point (p) and a line (A) that lies entirely within the plane, it must also contain all other points on that line (including all points on line B).\r<BR>\n" );
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document.write( "* **e. No other plane contains A ∪ B.**<BR>\n" );
document.write( "    * **Reason:** Two intersecting lines uniquely define a plane. \r<BR>\n" );
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document.write( "I hope this explanation is helpful! Let me know if you have any further questions.<BR>\n" );
document.write( " </SPAN>\n" );
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