document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209061>Question 1209061</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Verify that the points (0,0), (a,0) and (c,d) are the vertices of an equilateral triangle. \r<BR>\n" );
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document.write( "Note: c = (a/2) and d = (a•sqrt{3})/2\r<BR>\n" );
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document.write( "Then show that the midpoints of the 3 sides are the vertices of a second equilateral triangle.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848173 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848173\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Verify Equilateral Triangle**\r<BR>\n" );
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document.write( "* **Points:**<BR>\n" );
document.write( "    * A: (0, 0)<BR>\n" );
document.write( "    * B: (a, 0)<BR>\n" );
document.write( "    * C: (c, d) = (a/2, (a√3)/2)\r<BR>\n" );
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document.write( "* **Distances:**<BR>\n" );
document.write( "    * AB = √[(a - 0)² + (0 - 0)²] = √a² = a<BR>\n" );
document.write( "    * BC = √[(a/2 - a)² + ((a√3)/2 - 0)²] = √[(-a/2)² + (a√3/2)²] = √(a²/4 + 3a²/4) = √(4a²/4) = a<BR>\n" );
document.write( "    * AC = √[(a/2 - 0)² + ((a√3)/2 - 0)²] = √[(a/2)² + (a√3/2)²] = √(a²/4 + 3a²/4) = √(4a²/4) = a\r<BR>\n" );
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document.write( "Since AB = BC = AC = a, the points A, B, and C form an equilateral triangle.\r<BR>\n" );
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document.write( "**2. Find Midpoints**\r<BR>\n" );
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document.write( "* **Midpoint of AB (M1):** <BR>\n" );
document.write( "    * [(0 + a)/2, (0 + 0)/2] = (a/2, 0)\r<BR>\n" );
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document.write( "* **Midpoint of BC (M2):** <BR>\n" );
document.write( "    * [(a + a/2)/2, (0 + (a√3)/2)/2] = (3a/4, (a√3)/4)\r<BR>\n" );
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document.write( "* **Midpoint of AC (M3):** <BR>\n" );
document.write( "    * [(0 + a/2)/2, (0 + (a√3)/2)/2] = (a/4, (a√3)/4)\r<BR>\n" );
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document.write( "**3. Verify Equilateral Triangle for Midpoints**\r<BR>\n" );
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document.write( "* **M1M2:** <BR>\n" );
document.write( "    * √[((3a/4) - (a/2))² + ((a√3)/4 - 0)²] = √[(a/4)² + (a√3/4)²] = √(a²/16 + 3a²/16) = √(4a²/16) = a/2\r<BR>\n" );
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document.write( "* **M2M3:** <BR>\n" );
document.write( "    * √[((a/4) - (3a/4))² + ((a√3)/4 - (a√3)/4)²] = √[(-a/2)² + 0²] = a/2\r<BR>\n" );
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document.write( "* **M3M1:** <BR>\n" );
document.write( "    * √[((a/2) - (a/4))² + (0 - (a√3)/4)²] = √[(a/4)² + (a√3/4)²] = √(a²/16 + 3a²/16) = a/2\r<BR>\n" );
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document.write( "Since M1M2 = M2M3 = M3M1 = a/2, the midpoints M1, M2, and M3 also form an equilateral triangle.\r<BR>\n" );
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document.write( "**Therefore, it is proven that if the points (0, 0), (a, 0), and (c, d) (where c = a/2 and d = (a√3)/2) are the vertices of an equilateral triangle, then the midpoints of the sides of this triangle also form an equilateral triangle.**<BR>\n" );
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