A more thorough examination of similar triangles.
Using the same procedure and picture as before we will examine each individual similarity case.
J is the centroid of triangle BCF so BC/JB = square root of 3/1 and H is the centroid of triangle AEB so AB/HB=square root of 3/1. Thus, BC/JB = AB/ HB=square root of 3/1.
Angle EBH is equal to Angle JBC which equals 30 degrees because segment BH and BJ are angle bisectors of the respective equilateral triangles.
Using the angle addition postulate, Angle EBC is congruent to Angle HBJ.
Triangle EBC is similar to Triangle HBJ by the SAS Postulate for similar triangles.
Since corresponding sides of similar triangles are proportional
Again we repeat the above procedure...
I is the centroid of Triangle ADC so CD/CI = square root of 3/1 and J is the centroid of Triangle BCF so BC/JC=square root of 3/1. Thus, CD/CI = BC/ JC=square root of 3/1.
Angle DCI is equal to Angle JCB which equals 30 degrees because segment IC and JC are angle bisectors of the respective equilateral triangles.
Using the angle addition postulate, Angle JCI is congruent to Angle BCD.
Triangle JCI is similar to Triangle BCD by the SAS Postulate for similar triangles.
Since corresponding sides of similar triangles are proportional