Dissecting a Square into Acute Triangles

Date: 11/09/1999 at 11:26:31
From: Adam Sheraden
Subject: Geometry

Can you dissect a square into a finite number (fewer than 14) of acute 
triangles?

Date: 11/09/1999 at 13:07:06
From: Doctor Rob
Subject: Re: Geometry

Thanks for writing to Ask Dr. Math, Adam.

Put the vertices of the square at A(0,0), B(10,0), C(10,10), and 
D(0,10). Mark points along the sides at E(5,0), F(10,6), G(5,10), and 
H(0,6). Mark points in the interior at I(3,4), J(7,4), K(6,7), and 
L(4,7). Then connect AI, BJ, CK, DL, EI, EJ, FJ, FK, GKJ, GLI, HI, and 
HL. Prove that GKJ and GLI are straight lines. Then prove that the 12 
triangles formed are all acute.

For just 8 triangles, see the top of David Eppstein's page from the 
Geometry Junkyard, Triangles and Simplices:

   http://www1.ics.uci.edu/~eppstein/junkyard/triangulation.html

- Doctors Rob and Schwa, The Math Forum
  http://mathforum.org/dr.math/