# Picturing the Proof of the Coble construction

The following diagrams outline the geometric reasoning behind the Coble construction.

Begin with the original diagram and add several auxiliary circles and lines.  Note the three congruent rhombuses, BHCA, GBAD, and JGDI, with E at the intersection of the diagonals of the middle rhombus DGBA, which is known to be at the midpoint of each diagonal.  Note the similar triangles ACF and ICJ.
Since |CA| = 1, |CI| = 3, and |IJ| =1, ratios of similar triangles shows that |AF| = 1/3.

A referee suggested a nice alternative outline of a proof.  The triangles ADG and ABG are both equilateral triangles.  Thus, the measure of angle DAE equals the measure of angle BAE; by Side-Angle-Side, the triangles DAE and BAE are congruent.  Then the length of the segments DE and BE are equal, and also the length of the segments CA and AD are equal, so the segments AB and CE are medians of the triangle BCD.  It is now a well known fact that the intersection F of the medians is at the centroid which is one-third of the distance along the median AB.

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