# Golden Ratio In Right Isosceles Triangle

The following has been posted by Tran Quang Hung at the CutTheKnotMath facebook page.

Let $ABC\;$ be an Right Isosceles triangle, $D\;$ and $E\;$ the midpoints of the sides; $F\;$ be the foot of the latitude from $C;\;$ $G\;$ the intersection of $CF\;$ and circle $(ABED).$

Then $G\;$ divides $CF\;$ in the Golden Ratio: $\displaystyle\frac{FG}{CG}=\varphi.$

Assume $AB=4\;$ and expand the diagram as shown:

Then $AI=DI=1;\;$ $AD=\sqrt{2};\;$ $BI=3;\;$ $BD=\sqrt{10};\;$ area $S=[\Delta ABD]=2,\;$ and from $abc=4RS,$ the circumradius $R\;$ (i.e., the radius of $(ABED))$ is $\displaystyle R=\frac{\sqrt{2}\cdot 4\cdot\sqrt{10}}{4\cdot 2}=\sqrt{5}.$

By the Power of a Point Theorem, for $C,\;$ $AC\cdot CD=CG\cdot CH.\;$ With $x=CG,\;$ this becomes $4=x(x+2\sqrt{5}),\;$ so that $x=-\sqrt{5}+3.\;$ Since $CF=2,\;$ $\displaystyle\frac{FG}{CG}=\frac{\sqrt{5}-1}{3-\sqrt{5}}=\varphi.$

Grégoire Nicollier has observed that a shorter proof follows from the fact that the center of the circle lies $1\;$ unit below $F\;$ because this point is on equal distance $\sqrt{5}\;$ from $A,\;$ $B,\;$ $E,\;$ and $D.\;$ Then it is immediate that $FG=\sqrt(5)-1\;$ and $GC = 2-FG.$

### Golden Ratio

- Golden Ratio in Geometry
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- Golden Ratio in an Irregular Pentagon
- Golden Ratio in a Irregular Pentagon II
- Inflection Points of Fourth Degree Polynomials
- Wythoff's Nim
- Inscribing a regular pentagon in a circle - and proving it
- Cosine of 36 degrees
- Continued Fractions
- Golden Window
- Golden Ratio and the Egyptian Triangle
- Golden Ratio by Compass Only
- Golden Ratio with a Rusty Compass
- From Equilateral Triangle and Square to Golden Ratio
- Golden Ratio and Midpoints
- Golden Section in Two Equilateral Triangles
- Golden Section in Two Equilateral Triangles, II
- Golden Ratio is Irrational
- Triangles with Sides in Geometric Progression
- Golden Ratio in Hexagon
- Golden Ratio in Equilateral Triangles
- Golden Ratio in Square
- Golden Ratio via van Obel's Theorem
- Golden Ratio in Circle - in Droves
- From 3 to Golden Ratio in Semicircle
- Another Golden Ratio in Semicircle
- Golden Ratio in Two Squares
- Golden Ratio in Two Equilateral Triangles
- Golden Ratio As a Mathematical Morsel
- Golden Ratio in Inscribed Equilateral Triangles
- Golden Ratio in a Rhombus
- Golden Ratio in Five Steps
- Between a Cross and a Square
- Four Golden Circles
- Golden Ratio in Mixtilinear Circles
- Golden Ratio With Two Equal Circles And a Line
- Golden Ratio in a Chain of Polygons, So to Speak
- Golden Ratio With Two Unequal Circles And a Line
- Golden Ratio In a 3x3 Square
- Golden Ratio In a 3x3 Square II
- Golden Ratio In Three Tangent Circles
- Golden Ratio In Right Isosceles Triangle
- Golden Ratio Poster
- Golden Ratio Next to the Poster
- Golden Ratio In Rectangles
- Golden Ratio In a 2x2 Square: Without And Within
- Golden Ratio With Two Unequal Circles And a Line II
- Golden Ratio in Equilateral and Right Isosceles Triangles
- Golden Ratio in a Butterfly Astride an Equilateral Triangle
- The Golden Pentacross
- 5-Step Construction of the Golden Ratio, One of Many
- Golden Ratio in 5-gon and 6-gon
- Golden Ratio in an Isosceles Trapezoid with a 60 degrees Angle
- Golden Ratio in Pentagon And Two Squares
- Golden Ratio in Pentagon And Three Triangles
- Golden Ratio in a Mutually Beneficial Relationship
- Star, Six Pentagons and Golden Ratio
- Rotating Square in Search of the Golden Ratio
- Cultivating Regular Pentagons
- Golden Ratio in an Isosceles Trapezoid with a 60 degrees Angle II
- More of Gloden Ratio in Equilateral Triangles
- Golden Ratio in Three Regular Pentagons
- Golden Ratio in Three Regular Pentagons II
- Golden Ratio in Wu Xing
- Golden Ratio In Three Circles And Common Secant
- Flat Probabilities on a Sphere
- Golden Ratio in Square And Circles
- Golden Ratio in Square
- Golden Ratio in Two Squares, Or, Perhaps in Three
- Golden Ratio in Isosceles Triangle
- Golden Ratio in Circles
- Golden Ratio in Isosceles Triangle II
- Golden Ratio in Yin-Yang

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