Golden ratio in a circle
When studying constructions of golden ratio or dividing one segment by golden ratio we always see some lines, some circles and some segments. In this note we try to divide one radius segment of a circle by golden ratio using this circle, its radius and some other special lines. We also use the result to construct some new golden ratio configurations and explain essences some famous well known golden ratio configurations.
Naturally on real axis L with origin O we construct a unit circle (O) with diameter AA' here A, A' are on the axis L, so OA = +1, OA' = -1. Suppose X, Y are any two points on segment (-1, +1). Perpendiculars to L from X, Y intersect unit circle (O) at B, C respectively. Moreover B, C are in two different sides with respect to L. Point M is intersection of two lines BC, OA.
For convenient we denote x = OX, y = OY, m = OM are real coordinates of X, Y, M. (See figure 1).
Figure 1
We calculate m in terms of x, y
Two triangles MXB, MYC are similar, therefore:
(y – m)/(m – x) = CY/BX (1)
By Pythagorean theorem in two right triangles OXB, OYC:
BX = Sqrt(OB^2 – OX^2) = Sqrt(1 – x^2) (2)
CY = Sqrt(OC^2 – OY^2) = Sqrt(1 – y^2) (3)
Replacing (2), (3) in (1) and find m by x, y we have:
m = (x*Sqrt(1 – y^2) + y*Sqrt(1 – x^2))/(Sqrt(1 – x^2) + Sqrt(1 – y^2)) (4)
Now we want to find condition that M divides AO by golden ratio.
M divides AO by golden ratio if and only if:
AM/MO = φ if and only if m = 1/(1 + φ)
Solve equation (4) with m = 1/(1 + φ) we have found:
y = (3x – 2)/(2x – 3) (5)
here x is any real point on segment (-1,+1)
If use (5) to calculate x in term of y then
x = (3y – 2)/(2y – 3) (6)
It means x, y are symmetric each other in equation (5)
So we have a golden ratio in a circle theorem:
With construction of X, Y, M as in figure 1, AM/MO = φ if and only if
y = (3x – 2)/(2x – 3) (5) or
x = (3y – 2)/(2y – 3) (6) here -1< x, y < +1
If we take proper value x then M can be constructed by ruler and compass. Following are some interesting cases:
1. x = 0, y = 2/3
It is article "From 3 to Golden Ratio in Semicircle" in Cut The Knot:
http://www.cut-the-knot.org/do_you_know/BuiGoldenRatio.shtml
2. x = -1/2, y = 7/8
It is article "Another Golden Ratio in Semicircle" in Cut The Knot:
http://www.cut-the-knot.org/do_you_know/BuiGoldenRatio2.shtml
3. x = 1/4, y = 1/2
It is Kurt Hofstetter "Another 5-step division of a segment in the golden section" Forum Geometricorum 4 (2004) 21–22 (See figure 2):
http://forumgeom.fau.edu/FG2004volume4/FG200402index.html
Figure 2
Interesting remark:
The case (2) can be used to create 6-step division of a segment in the golden ratio as following (See figure 3):
Figure 3
We need to divide segment OA by golden ratio
1. To construct a circle centered at O passing through A.
2. To construct a circle centered at A passing through O.
These two circle intersect at G, H
3. To construct line GH. GH intersects OA at I.
4. To construct a circle centered at A passing through I.
This circle intersects circle (O) at C
5. To construct a circle centered at G passing through A.
Other than A, this circle intersects circle (O) at B
6. To construct line BC.
BC divide OA by golden ratio at M.
Please note that this 6-step division configuration is very closed with Kurt Hofstetter "A 5 -step Division of a Segment in the Golden Section" Forum Geometricorum, 3 (2003) 205--206:
http://forumgeom.fau.edu/FG2003volume3/FG200322index.html
You may be can find another x, y with another interesting constructions.
