# Golden Ratio as a Mathematical Morsel

Here's another construction of the Golden Ratio posted by Tran Quang Hung at the CutTheKnotMath facebook page.

Square $MNPQ$ is inscribed into square $ABCD$ so that the lines joining their vertices intersect the sides of $MNPQ$ at the midpoints $X,Y,Z,W,$ as shown:

Prove that the vertices of $MNPQ$ divide the sides of $ABCD$ in the Golden Ratio, e.g. $\displaystyle\frac{BM}{AM}=\phi.$

### Construction

This is how square $MNPQ$ can be inscribed into square $ABCD$ as required in the statement. The construction is backwards, starting with square $MNPQ.$ Erect semicircles on its sides and draw the lines through the vertices and the midpoints, e.g., $MZ,$ till they intersect the semicircles:

The intersections form square $ABCD.$

### Proof

I find it convenient to add a third square - $FGHI$ - to the diagram, and designate the small one that is already present as $RSTU:$

These squares made an appearance early at the site development as an example of a simple math curiosity - mathematical morsels, as I referred to them at the time. $FGHI$ is being cut into 9 copies of $RSTU.$

Let the side of the smallest square be $2:$ $RS=\ldots =2.$ Then $MS=4,$ $SN=2,$ and $MN=2\sqrt{5}.$ Since $Y$ is the circumcenter of the right triangle $MNB,$ $BY=\sqrt{5},$ so that $BT=\sqrt{5}+1.$ In triangle $BTM,$

$BM^{2}=(\sqrt{5}+1)^{2}+2^{2}=2\sqrt{5}(\sqrt{5}+1).$

Similarly, in triangle $AMJ,$ $MJ=2,$ $AJ=\sqrt{5}-1,$ such that

$AM^{2}=(\sqrt{5}-1)^{2}+2^{2}=2\sqrt{5}(\sqrt{5}-1).$

Thus we get

$\displaystyle\frac{BM^{2}}{AM^{2}}=\frac{\sqrt{5}+1}{\sqrt{5}-1}=\frac{\sqrt{5}+3}{2}=1+\phi=\phi^{2},$

just as required.

### 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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