Geometric constructions of Phi in Circles

Phi appears in a number of geometric constructions using circles

There are a number of geometric constructions using a circle which produce phi relationships, as described below.

Among mathematicians, there's a bit of a competition to see how few lines can be used to create a phi proportion, or golden section, in the construction, or how many golden sections can be created with the least number of lines.

Three circle construction

Put three circles with a diameter of 1 (AB and DE) side by side and construct a triangle that connects the bottoms of the outside circles (AC) and the top and bottom of the outside circles (BC).  The dimensions are as follows:

AB = 1

AC = 2

BC = √5

DE = 1

The line BC thus expresses the following embedded phi relationships:

BE = DC = (√5-1)/2+1  = (√5+1)/2 = 1.618 ... = Phi

BD = EC = (√5-1)/2 = 0.618... = phi

This simple and elegant way of expressing the most standard mathematical expression of Phi was discovered and contributed by Bengt Erik Erlandsen on 1/11/2006.

Equilateral triangle construction

Insert an equilateral triangle DEF inside a circle.  Find the midpoints of each leg at ABC.

The ratio of the length of segment AG to segment AB is Phi, or 1.618 0339 887 ...

This construction was developed by George Odom and published in American Mathematics Monthly, 90 (1983) 482, with the solution in 93 (1986) 572.

Enhanced equilateral triangle construction

Here's a very interesting enhancement to the basic equilateral triangle construction above:

Connect the points with lines at AF and DG (in red).

At Y, the intersection of DG and EF, create perpendicular lines from Y to AF at Z, and again from Y to ED at W.

This produces a number of phi relationships, or golden sections:

Can you find more phi relationships?  If so send them in!

This construction, while similar to the Odom construction, was developed independently by Hans J. Dettmer as an elegant solution to dividing a prism in equal volumes, as described in the attached paper.

Concentric circle construction

Here's a construction using three concentric circles whose radiuses are in a ratio of 1 : 2 : 4.

Draw a tangent from the small circle through the other two, crossing points A and B and extending to G.

The ratio of the length of segment AG to segment AB is Phi, or 1.618 0339 887 ...

Proof:  AB = 2 * 3^½ and AG = 15^½ + 3^½, which by factoring out the 3^½ can be reduced to a ratio of 2 to (5^½+1), or Phi.

This construction was developed by Sam Kutler and submitted by Steve Lautizar.

Overlapping circles construction

This construction can be created by simply drawing five circular arcs.

Construct concentric circles of radius 1 and 2 with a center point at C.

Construct concentric circles of radius 1 and 2 with a center point at D.

Draw a line from the intersection points of the two smaller circles at A
to the intersection point of the two larger circles at G.

The ratio of the length of segment AG to segment AB is Phi, or 1.618 0339 887 ...

Proof:  AB/AG = ( 2 Ö 3 ) / ( Ö15 + Ö3)  =  2 / ( Ö5 = Ö1)  =  2 / ( Ö5 = Ö1)  = Phi

This construction was developed by Kurt Hotstetter in 2002 and published in Forum Geometricorum, Volume 2 (2002) 65-66.