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.[br]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].[br]2. Given any triangle, ABC, how can the point, O, be found (using a straightedge and compass)?[br]3. How can all the coloured triangles be constructed?[br]4. What if one of the angles of ABC measures more than 120[math]^\circ[/math]?