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Show that F, D, H, G are concyclic.
diagram: http://imgur.com/a/5gmA1
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From this Wikipedia article:
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle.
This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
The center of the circle and its radius are called the circumcenter and the circumradius respectively.
Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral.
Solution to the Problem
Make a sketch (you can use the referenced plot as a starting Figure).
Connect the points F and G by the straight line segment FG.
Connect the points F and D by the straight line segment FD.
Connect the points G and H by the straight line segment GH.
The point I is the intersection of the circles and the intersection of the straight lines FH and DG at the same time.
(The point I is shown in the Figure, but I specially turn on your attention)
Notice that the angle IHG is a right angle, since it leans on the diameter IG of the circle A.
Therefore, the angle FHG is a right angle, too.
Similarly, the angle FDI is a right angle, since it leans on the diameter FI of the circle O.
Therefore, the angle FDG is a right angle, too.
Thus we have two right angles, FHG and FDG, that are leaning on the same segment FG.
It means that the points D and H lie on the circle having FG as the diameter.
In turn, it means that the points F, G, H and D are concyclic.
The proof is completed.
I used the lemmas that are well known in systematic course of Elementary Plane Geometry:
Lemma 1. An inscribed angle leaning on the diameter of a circle is a right angle.
Lemma 2. If an inscribed angle is a right angle, then it leans on the diameter of the circle.
Regarding these lemmas, see the lesson
An inscribed angle in a circle
in this site.
Solved.
I'd like to inform you that there is this free of charge online textbook on Geometry
GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.
The lesson I referred above is part of this online textbook in the section/topic Properties of circles, their chords, secants and tangents.
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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 ?
Thank you.