document.write( "Question 1011877: A is a point where the circle with equation x^2 +y^2 = 16 cuts the x axis. Find the locus of the midpoints of all chords of this circle that contain A.\r
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Algebra.Com's Answer #627739 by ikleyn(52788)\"\" \"About 
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document.write( "You just know that your original figure is a circle of the radius 4 with the center at the origin \r\n" );
document.write( "of the coordinate system of the coordinate plane.\r\n" );
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document.write( "You also know that two yours basic points are the endpoints of the diameter of the circle, \r\n" );
document.write( "which lies on the horizontal axis y = 0. These points are A1 = (-4,0) and A2 = (4,0).\r\n" );
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document.write( "Now, the answer is:\r\n" );
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document.write( "    The locus of the midpoints of all chords of this circle that contain A1 is the circle of the radius 2 \r\n" );
document.write( "    with the center at the point C1 = (-2,0). \r\n" );
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document.write( "        It is the circle of the radius half of the radius of the original circle and touching \r\n" );
document.write( "        the original circle from the interior at the point A1 (shown in red in the Figure 1)\r\n" );
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document.write( "    The locus of the midpoints of all chords of this circle that contain A2 is the circle of the radius 2 \r\n" );
document.write( "    with the center at the point C2 = (2,0). \r\n" );
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document.write( "        It is the circle of the radius half of the radius of the original circle and touching \r\n" );
document.write( "        the original circle from the interior at the point A2 (shown in green in the Figure 1).\r\n" );
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document.write( "             Figure 1\r\n" );
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document.write( "                 Figure 2\r\n" );
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Proof\r\n" );
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document.write( "Let A be an arbitrary, but fixed point of the circle O (Figure 2).\r\n" );
document.write( "Let AB be some chord in the given circle with the points A and B on the circle (Figure 2). \r\n" );
document.write( "Let D be the middle point of the chord AB. Let the point C bisects the radius OA.\r\n" );
document.write( "We need and we are going to prove that the length of the interval CD is half of the length of the radius OA.\r\n" );
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document.write( "Connect the center of the circle O with the point D.\r\n" );
document.write( "Then OD is the median in the triangle AOB.\r\n" );
document.write( "This triangle is isosceles (OA and OB are congruent as the radii of the circle).\r\n" );
document.write( "In the isosceles triangle the median to the base coincides with the altitude (see the lesson \r\n" );
document.write( "  An altitude a median and an angle bisector in the isosceles triangle in this site).\r\n" );
document.write( "So, OD is perpendicular to AB and the triangle ADO is right-angled triangle with the right angle at the vertex D.\r\n" );
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document.write( "Now, CD is the median in the right-angled triangle ADO drawn to its hypotenuse.\r\n" );
document.write( "In the right-angled triangle the median drawn to hypotenuse has the measure half of the hypotenuse\r\n" );
document.write( "  (see the lesson Median drawn to the hypotenuse of a right triangle in this site).\r\n" );
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document.write( "Thus we proved our statement: the distance from the point C to the middle of the chord AB is half the radius of the \r\n" );
document.write( "original circle. In other words, middle points of all chords passing through the point A are equidistant from the point C.\r\n" );
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