# Golden Ratio In a 3x3 Square II

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

In a $3\times 3\;$ let $P\;$ be the incenter of the inscribed isosceles triangle as shown below; $S\;$ and $T\;$ the two opposite corners of the middle square:

.

Then $P\;$ divides $ST\;$ in the Golden Ratio: $\displaystyle\frac{PT}{PS}=\varphi.$

Assuming the side length of the small squares is $1,\;$ the two sides of the triangle are equal to $\sqrt{10}\;$ while its base is $2\sqrt{2},\;$ making its semiperimeter $p=\sqrt{10}+\sqrt{2}.\;$ Its area is obviously $S=\frac{1}{2}2\sqrt{2}\cdot 2\sqrt{2}=4.\;$ We may thus compute its inradius: $\displaystyle r=PT=\frac{S}{p}=\frac{2\sqrt{2}}{\sqrt{5}+1}.\;$ Further, $\displaystyle PS=\sqrt{2}-r=\sqrt{2}\frac{\sqrt{5}-1}{\sqrt{5}+1}\;$ so that, finally,

$\displaystyle\frac{PT}{PS}=\frac{2\sqrt{2}}{\sqrt{5}+1}\cdot \frac{1}{\sqrt{2}}\cdot \frac{\sqrt{5}+1}{\sqrt{5}-1}=\frac{2}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{2}.$

Here's an alternative proof by Quang Duong. Let $D\;$ be the incenter of $\Delta ABT:$

Apply Menelaus' Theorem in $\Delta BPS\;$ with the transversal $GDT:$ $\displaystyle\frac{ST}{TP}\cdot\frac{PD}{DB}\cdot\frac{BG}{GS}=1\;$ which reduces to $\displaystyle\frac{ST}{TP}=\frac{DB}{PD}.$

Since $D\;$ is the incenter of $\Delta ABT,\;$ $\displaystyle\frac{DB}{DP}=\frac{AB+BT}{AT}.\;$ But $AB=\sqrt{10},\;$ $BT=\sqrt{2},\;$ $AT=2\sqrt{2}.\;$ Thus, $\displaystyle\frac{DB}{PD}=\frac{AB+BT}{AT}=\frac{\sqrt{5}+1}{2}\;$ and, consequently, $\displaystyle\frac{ST}{TP}=\frac{\sqrt{5}+1}{2}\;$ from which, too, $\displaystyle\frac{PT}{SP}=\frac{\sqrt{5}+1}{2}.$

Grégoire Nicollier has pointed to the shortest proof: by the Angle Bisector theorem, $AP:PT=\sqrt{10}:\sqrt{2},\;$ and the result follows from here.

### Golden Ratio

- Golden Ratio in Geometry
- Golden Ratio in Regular Pentagon
- 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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