Trapped! (Posted on 2004-02-19)

	Submitted by DJ
Rating: 4.4444 (9 votes)
Solution:	(Hide)
Draw quadrilateral ABCD. If the convex hull is a triangle, then the fourth point lies inside its circumcircle. So suppose ABCD is convex. One pair of opposite angles must have sum greater than 180° (otherwise the points would lie on a circle; if we take four points on a circle, opposing angles always sum 180°). Suppose they are A and C. Then we claim that C lies inside the circle through A, B, D. Take a point C' on the ray DC on the far side of C from D such that: m Then, m So, C' lies on the circle. Hence C, which lies on the segment C'D, lies inside the circle.

	Subject	Author	Date
	Question on the official solution	JayDeeKay	2010-03-05 10:52:09
	re(2): 'Within' includes 'on'?	TomM	2004-02-20 22:21:43
	Four Points in a Plane	Anthony	2004-02-20 21:11:38
	re: a proof	TomM	2004-02-20 20:24:59
	No Subject	tan	2004-02-20 00:51:12
	Euler, Feuerbach, et. al.	Penny	2004-02-19 18:59:18
	re: 'Within' includes 'on'?	Federico Kereki	2004-02-19 15:09:40
	'Within' includes 'on'?	e.g.	2004-02-19 14:31:20
	a proof	Charlie	2004-02-19 14:17:56