CONCEPT 1 - Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

1. The construction of an inscribed equilateral.

So why does this work? 

2. The construction of an inscribed square.

So why does this work? 

Reason #1 – A Square has diagonals that are perpendicular and congruent.  The perpendicular diameters determine the square.

Reason #2 – The perpendicular bisectors form a central angle of 90° which divide the circle into 4 congruent parts, thus forming the square.

3. The construction of an inscribed hexagon.

So why does this work? 

Reason #1 – Step (D) divided the circle into six congruent arcs, thus six congruent chords.

Reason #2 – Step (D) created six equilateral triangles, ΔFAB. ΔBAG, ΔGAE, ΔEAC, ΔCAD, and ΔDAF) dividing the circle into six congruent parts.