document.write( "Question 1048171: Show that F, D, H, G are concyclic. \r
\n" ); document.write( "\n" ); document.write( "diagram: http://imgur.com/a/5gmA1\r
\n" ); document.write( "\n" ); document.write( "I don't know how to prove that the points are concyclic. Could you please help?
\n" ); document.write( "Thank you.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #663996 by ikleyn(53751) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Show that F, D, H, G are concyclic. \r
\n" ); document.write( "\n" ); document.write( "diagram: http://imgur.com/a/5gmA1
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this Wikipedia article:\r
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "     In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. \r\n" );
document.write( "     This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. \r\n" );
document.write( "     The center of the circle and its radius are called the circumcenter and the circumradius respectively. \r\n" );
document.write( "     Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solution to the Problem\r
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Make a sketch (you can use the referenced plot as a starting Figure).\r\n" );
document.write( "\r\n" );
document.write( "Connect the points F and G by the straight line segment FG.\r\n" );
document.write( "Connect the points F and D by the straight line segment FD.\r\n" );
document.write( "Connect the points G and H by the straight line segment GH.\r\n" );
document.write( "\r\n" );
document.write( "The point I is the intersection of the circles and the intersection of the straight lines FH and DG at the same time. \r\n" );
document.write( "    (The point I is shown in the Figure, but I specially turn on your attention)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Notice that the angle IHG is a right angle, since it leans on the diameter IG of the circle A.\r\n" );
document.write( "\r\n" );
document.write( "Therefore, the angle FHG is a right angle, too.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Similarly,  the angle FDI is a right angle, since it leans on the diameter FI of the circle O.\r\n" );
document.write( "\r\n" );
document.write( "Therefore, the angle FDG is a right angle, too.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus we have two right angles, FHG and FDG, that are leaning on the same segment FG.\r\n" );
document.write( "\r\n" );
document.write( "It means that the points D and H lie on the circle having FG as the diameter.\r\n" );
document.write( "\r\n" );
document.write( "In turn, it means that the points F, G, H and D are concyclic.\r\n" );
document.write( "\r\n" );
document.write( "The proof is completed.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "I used the lemmas that are well known in systematic course of Elementary Plane Geometry:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Lemma 1. An inscribed angle leaning on the diameter of a circle is a right angle.\r\n" );
document.write( "\r\n" );
document.write( "Lemma 2. If an inscribed angle is a right angle, then it leans on the diameter of the circle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Regarding these lemmas, see the lesson\r\n" );
document.write( "    An inscribed angle in a circle\r\n" );
document.write( "in this site.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I'd like to inform you that there is this free of charge online textbook on Geometry\r
\n" ); document.write( "\n" ); document.write( " GEOMETRY - YOUR ONLINE TEXTBOOK \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The lesson I referred above is part of this online textbook in the section/topic Properties of circles, their chords, secants and tangents.\r
\n" ); document.write( "\n" ); document.write( "---------------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "May I ask you please to send me the name of the source to this problem (textbook, book?) through the \"Thank you\" message window/form ?\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Thank you.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );