document.write( "Question 55924: draw a big circle 1 and draw another smaller circle 2 to the right of the big circle 1 and touches the big circle 1. Draw a line that tangents to both circles at the bottom. Draw a third smaller circle 3 in the space between the two larger circles and the line so that the third circle is touching the two bigger circles and tangent to the line. r1, r2, and r3 are radii of circle 1, circle 2, and circle 3. Express r3 in terms of r1 and r2. \n" ); document.write( "
Algebra.Com's Answer #37943 by Edwin McCravy(20054)\"\" \"About 
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draw a big circle 1 and draw another smaller circle 2\r\n" );
document.write( "to the right of the big circle 1 and touches the big \r\n" );
document.write( "circle 1. Draw a line that tangents to both circles \r\n" );
document.write( "at the bottom. Draw a third smaller circle 3 in the \r\n" );
document.write( "space between the two larger circles and the line so \r\n" );
document.write( "that the third circle is touching the two bigger \r\n" );
document.write( "circles and tangent to the line. r1, r2, and r3 are \r\n" );
document.write( "radii of circle 1, circle 2, and circle 3. Express \r\n" );
document.write( "r3 in terms of r1 and r2.\r\n" );
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document.write( "Let A be the center of the circle 1, B the center of \r\n" );
document.write( "circle 2, and C the center of circle 3.\r\n" );
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document.write( "Draw triangle ABC.  \r\n" );
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document.write( "Draw the radii AD, CE and BF perpendicular to the \r\n" );
document.write( "tangent line, of which DE and EF are segments.\r\n" );
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document.write( "Draw a line thru C parallel to the tangent line.  \r\n" );
document.write( "Let it intersect AD at G and BF at H.\r\n" );
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document.write( "AG = r1-r3, BH = r2-r3\r\n" );
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document.write( "AC = r1+r3, BC = r2+r3\r\n" );
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document.write( "Triangles AGC and BHC are right triangles, so we \r\n" );
document.write( "can use the Pythagorean theorem:\r\n" );
document.write( "      _______    _________________    _____     ____\r\n" );
document.write( "GC = ÖAC²-AG² = Ö(r1+r3)²-(r1-r3)² = Ö4r1r3 = 2Ör1r3       \r\n" );
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document.write( "HC = ÖBC²-BH² = Ö(r2+r3)²-(r2-r3)² = Ö4r2r3 = 2Ör2r3\r\n" );
document.write( "                   ____     ____     __  __    __\r\n" );
document.write( "Now GH = GC+HC = 2Ör1r3 + 2Ör2r3 = 2Ör3(Ör1 + Ör2)\r\n" );
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document.write( "Draw a line thru B parallel to GH and DF. Let it \r\n" );
document.write( "intersect AD at I. \r\n" );
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document.write( "AB = r1+r2,  AI = r1-r2\r\n" );
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document.write( "Triangle AIB is a right triangle,  so we can use \r\n" );
document.write( "the Pythagorean theorem:\r\n" );
document.write( "      _______    _________________    _____     ____  \r\n" );
document.write( "IB = ÖAB²-AI² = Ö(r1+r2)²-(r1-r2)² = Ö4r1r2 = 2Ör1r2\r\n" );
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document.write( "Now IB = GH so\r\n" );
document.write( "  ____     __  __    __   \r\n" );
document.write( "2Ör1r2 = 2Ör3(Ör1 + Ör2)\r\n" );
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document.write( "Divide through by 2\r\n" );
document.write( "  ____    __  __    __   \r\n" );
document.write( " Ör1r2 = Ör3(Ör1 + Ör2)\r\n" );
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document.write( "Square both sides:\r\n" );
document.write( "           __    __  \r\n" );
document.write( "r1r2 = r3(Ör1 + Ör2)²\r\n" );
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document.write( "        r1r2    \r\n" );
document.write( "r3 = ------------\r\n" );
document.write( "     (Ör1 + Ör2)²\r\n" );
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document.write( "Edwin

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