Crop Circles

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Tangents of Crop Circles
Screen shot 2013-06-11 at 2.22.20 PM.png
Field: Geometry
Image Created By: Eiman Eltigani

Tangents of Crop Circles

Crop circles, formed by crushed crops, are a pattern of geometric shapes, such as triangles, circles, etc. They illustrate many geometric theorems and relationships between the shapes of the pattern.

Basic Description

Comparing three crop circles that aren’t exactly touching can form three tangent lines, with each line adjacent to all three of the circles. Connecting the center points of all the circles creates a triangle, which is equilateral. The circumscribed circle of the triangle includes all the center points of the three circles and shows relations of diameter to these circles.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Geometry

The ratio of the diameter of the triangle's circumscribed circle to the diameter of the circles at ea [...]

The ratio of the diameter of the triangle's circumscribed circle to the diameter of the circles at each corner is 4:3.

Screen shot 2013-06-18 at 10.55.55 AM.png

Line k is a tangent touching the edge of all three congruent circles. Line j and l pass through the center of the green circles. The distance from the center of a circle to a point on its edge creates the radius, and all radii (from congruent circles) are also congruent, so the distance between line j and k is congruent to distance between k and l, which measures a.

Screen shot 2013-06-19 at 10.42.06 AM.png

Triangle ABE is similar to triangle ACD since it is formed by a parallel line passing through triangle ACD. Line segment AB equals the distance of a, while line AC equals 2a, making the ratio of the two triangles 2:1. Therefore, 1/2BE = CD.

Screen shot 2013-06-19 at 10.43.36 AM.png

Angle <EBF = 90 and <DCF = 90 because they are angles formed by perpendicular bisectors. <BFE is congruent to <CFD because they are vertical angles bisected in half. Triangle BFE is similar to triangle CFD by Angle-Angle Postulate. Proven earlier, the ratio of BE and CD is 2:1, making the ratio of triangle BFE and CFD also 2:1. Since BF and FC share a distance of a, then FC equals 2/3a.

Screen shot 2013-06-19 at 10.44.34 AM.png

Line CD bisects line FG in half, so FC = CG. Since, FC = 2/3a, CG also equals 2/3a.

Screen shot 2013-06-18 at 11.05.33 AM.png

The total diameter of the circumscribed circle is 2 2/3a. The diameter of each of the original triangles equals 2a (2 radii). The ratio is 2 2/3a : 2a --> 4 : 3.

Why It's Interesting

Crop circles are interesting because it is an art, but also incorporates geometry into it. It's also a mystery because people have been trying to explain what creates them and many theories have come out of that from human creativity to flying UFOs.

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About the Creator of this Image

Eiman Eltigani, high school student at J.R. Masterman.

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