Castillon's Problem: Inscribe in a circle a triangle whose sides pass through three given points. This is Problem #29 of Heinrich Dörrie's [i]100 Great Problems of Elementary Mathematics.[/i] The following constructions and theorems are used: [b]The power of a circle at a point:[/b] [url]http://www.geogebratube.org/material/show/id/28302[/url] [b]Chord from a point and angle:[/b] [url]http://www.geogebratube.org/material/show/id/30756[/url]
Notes: -I understand [i]Sides of the triangle[/i] to mean the straight lines through the three points of the triangle. -The proof is pretty, but the relationships among the auxiliary points must be understood in advance of the construction. *This construction is not reliable with C outside the circle. The problem is better described by swinging a line from side to side: Solution 2, Steiner: [url]http://www.geogebratube.org/material/show/id/55007[/url]