<p>just realized that OA = OB because all radii congurent…but why does CA = CB?</p>
<p>CA = CB is given, and OA and OB are radii of the same circle.</p>
<p>the problem only gives that ARC CA = ARC CB</p>
<p><BOC is congruent to <AOC, therefore BOC and AOC are congruent by SAS, therefore BC = AC.</p>
<p>lol i just realized, congurent arcs have congruent chords :)</p>
<p>From which book was this exercise scanned? By the standards of US high school instruction it is a difficult and non-standard exercise.</p>
<p>@siserune: this is the book - [Amazon.com:</a> Geometry (9780395977279): Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen: Books](<a href=“http://www.amazon.com/dp/0395977274/?tag=tbook-20]Amazon.com:”>http://www.amazon.com/dp/0395977274/?tag=tbook-20)</p>
<p>That’s a standard textbook for an honors geometry course, well it’s the one I used when I was in 9th grade :P</p>
<p>Thanks for the reference. The book as a whole appears closer to the standard of US mass market geometry textbooks, but the specific problem excerpted is difficult for that level. Usually when a geometry text has problems requiring that level of sophistication they are more theoretical than computational (in US terms, USAMO-style proofs rather than AIME/AMC style numerical calculations), but the publishers probably assume that the target audience of high school teachers and students is more comfortable with concrete numbers in the exercises.</p>
<p>interesting probblem</p>
<p>I think one of the best book from Geometry was written by Prasolov. There are lots of olympic level problems divided into several categories. Here it is:</p>
<p>[Prasolov</a> book](<a href=“http://michaj.home.staszic.waw.pl/prasolov.html]Prasolov”>Prasolov book)</p>