Editing Golden ratio (section)

=====Golden triangle and golden gnomon=====
{{main|Golden triangle (mathematics)}}
[[File:Golden triangle (math).svg|235px|right|thumb|A [[Golden triangle (mathematics)|golden triangle]] {{mvar|ABC}} can be subdivided by an angle bisector into a smaller golden triangle {{mvar|CXB}} and a golden gnomon {{mvar|XAC}}.]]
The triangle formed by two diagonals and a side of a regular pentagon is called a ''golden triangle'' or ''sublime triangle''. It is an acute [[isosceles triangle]] with apex angle {{tmath|36^\circ}} and base angles {{tmath|72^\circ\!}}.<ref name=fletcher /> Its two equal sides are in the golden ratio to its base.<ref name=loeb /> The triangle formed by two sides and a diagonal of a regular pentagon is called a ''golden gnomon''. It is an obtuse isosceles triangle with apex angle {{tmath|108^\circ}} and base angle {{tmath|36^\circ\!}}. Its base is in the golden ratio to its two equal sides.<ref name=loeb /> The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a [[Pentagram|regular pentagram]] are golden triangles,<ref name=loeb /> as are the ten triangles formed by connecting the vertices of a [[regular decagon]] to its center point.<ref name=miller />

Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.<ref name=loeb />

If the apex angle of the golden gnomon is [[Angle trisection|trisected]], the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.<ref name=loeb />