Points A, B, C, and D (no three of which are collinear) lie in a plane. If point D lies randomly within ΔABC's anticomplementary triangle, then what is the probability that 4-gon ABCD is (concave, convex, reflex)?

What would be the answer if ΔABC is equilateral and "ΔABC's anticomplementary triangle" is replaced with "ΔABC's circumcircle"?

NOTE: The anticomplementary triangle of a given triangle is formed by three lines. Each line passes through a vertex of the given triangle and is parallel to the opposite side.

NOTE: The circumcircle of a given triangle is the unique circle which passes through each of its vertices.

Call the vertex farthest from A of the anticomplementary triangle A'.  Similarly for B' and C'.

Besides the triangle A'B'C', three smaller triangles ABC', AB'C and A'BC are also formed which are congruent to the original.

So A'B'C' is divided into 4 equal regions. 
If D is inside ABC the quadrilateral is concave: P(concave)=1/4
If D is inside AB'C the quad. is convex: P(convex)=1/4
If D is inside ABC' or A'BC it is reflex: P(reflex)=1/2

Posted by Jer on 2006-11-03 11:37:43