This applet illustrates Weitzenboeck's inequality (1919): [math]a^2+b^2+c^2\geq 4\sqrt{3}\triangle[/math] where [math]\triangle[/math] is the area of a triangle having sides of lengths, [math]a[/math], [math]b[/math] & [math]c[/math].
The applet shows any triangle, ABC. On each side is constructed an equilateral triangle, each of which is subdivided into three coloured triangles and one blank triangle. A point, O, is chosen in the interior of ABC such that |[math]\angle[/math]AOB|=|[math]\angle[/math]BOC|=|[math]\angle[/math]COA|=120[math]^\circ[/math]. Joining OA, OB & OC subdivides ABC into three triangles, one red, one blue and one green. 1. Explore the applet and suggest how it might be used to prove Weitzenboeck's inequality. Note, in particular, what happens if ABC is equilateral. Hint: the area of an equilateral triangle with side of length [math]s[/math] is [math]s^2 \frac{\sqrt{3}}{4}[/math]. 2. Given any triangle, ABC, how can the point, O, be found (using a straightedge and compass)? 3. How can all the coloured triangles be constructed? 4. What if one of the angles of ABC measures more than 120[math]^\circ[/math]?