A [i]golden triangle[/i] is an isosceles triangle in which the ratio of one of the equal sides to the base is the golden ratio, that is, [math]\varphi=(1+\sqrt{5})/2 \approx1.618[/math]. The angle at the vertex of the golden triangle is [math]36^ \circ[/math].[br][br]In this applet we start with an isosceles triangle whose equal sides have length [math]1[/math] and the base is equal to [math]\varphi[/math]. The angles at the base are [math]36 ^\circ[/math]. [br][list][br][*]Click the checkbox to show the first golden triangle.[br][/list][br]Next, we generate a sequence of smaller embedded golden triangles. This sequence is constructed by rotating the previous triangle on [math]72 ^\circ[/math] or [math]-72^ \circ[/math] and rescaling it with factor [math]1/ \varphi[/math].[br][list][br][*]Click the Construction ON/OFF button or drag the slider.[br][/list][br]Using the construction below, we can evaluate the series[br][math]S_1 = 1+1/ \varphi ^2+1/ \varphi ^4+....[/math], the series [math]S_2 = 1/ \varphi + 1/ \varphi ^3+....[/math] and also the series [math]S_3 = 1/ \varphi +1/ \varphi ^2+ 1/ \varphi ^3+...[/math][br][br]This applet is based on the note [i]Proof Without Words: An Infinite Series Using Golden Triangles[/i] by Steven Edwards, The College Mathematics Journal Vol. 45, No. 2 (March 2014), p. 120.