Golden Ratio in an Irregular Pentagon, Construction II
Let A, B, C be three noncollinear points and K complete a parallelogram ABCK. Extend AK to AD such that AD/AK = φ, the golden ratio. Similarly, extend CK to CE such that CE/CK = φ. Pentagon ABCDE is of a very peculiar kind.
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Explanation
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Copyright © 1996-2012 Alexander Bogomolny
Golden Ratio in an Irregular Pentagon, Construction II
Pentagon ABCDE has the diagonals parallel to the sides. In addition, the points of intersection of the diagonals divide each in the golden ratio.
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AD||BC and CE||AB by construction. Also, since AK/DK = φ = CK/EK, AC||AD. (AD/AK = φ implies AK/DK = φ because φ - 1 = 1/φ.)
Further, in ΔABD, KM||AB implying BD/BM = AD/AK = φ. Similarly, in ΔBCE, KL||BC implies BE/BL = φ. In particular, BD/BM = BE/BL so that LM||DE and DE/LM = φ.
In trapezoid DELM, DE/LM = φ and triangles DEK and LMK are similar so that DK/KL = EK/KM = φ.
No consider the four collinear points A, L, K, and D. We have,
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(AL + KL) / DK = φ = DK/KL,
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from which we derive
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φ = (DK + KL) / AL = DL/AL.
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But, as we already saw, BE/BL = φ and then also BL/EL = φ. Thus in the quadrilateral ABDE the diagonals BE and AD divide each other in equal ratios: BL/EL = DL/AL (= φ).
This tells us that ABDE is a trapezoid with BD||AE. Similarly, CD||BE.
Now, points K and L divide AD in the golden ration. Points K and M divide in the golden ration diagonal CE. It's now not difficult to verify that this is also true of points L, M, N, P and the remaining diagonals.
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- Take-Away Games>
- Number 8 Is Interesting
- Curry's Paradox
- A Problem in Checker-Jumping
- Fibonacci's Quickies
- Fibonacci Numbers in Equilateral Triangle
Golden Ratio
- Golden Ratio in Geometry
- 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
|Activities|
|Contact|
|Front page|
|Contents|
|Geometry|
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Copyright © 1996-2012 Alexander Bogomolny
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