
Remember  use your compass
and straightedge only! 

Regular Hexagon
(inscribed in a circle) 


A regular hexagon is a sixsided figure in which all of its angles are congruent and all of its sides are congruent.

Given: a piece of paper
Construct: a regular hexagon inscribed in a circle 

STEPS:
1. Place your compass point on the paper and draw a circle. (Keep this compass span!)
2. Place a dot, labeled P, anywhere on the circumference of the circle to act as a starting point.
3. Without changing the span on the compass, place the compass point on P and swing a small arc crossing the circumference of the circle.
4. Without changing the span on the compass, move the compass point to the intersection of the previous arc and the circumference and make another small arc on the circumference of the circle.
5. Keep repeating this process of "stepping" around the circle until you return to point P.
6. Starting at P, connect to each arc on the circle forming the regular hexagon.

Full circle view:


Proof of Construction: PA = AB = BC = CD = DE since these lengths represent copies of the radius of circle O. But how do we know for sure that the last length, EP, coincided exactly with point P? Is EP actually the same length as the other copied radii?
ΔDOE is an equilateral triangle since it has 3 sides of equal length (DO and OE are radii lengths and DE is a copy of this radii length). In the same manner, ΔCOD, ΔBOC. ΔAOB and ΔPOA are also equilateral triangles. Since the interior angles of an equilateral triangle each contain 60º, m∠COD = m∠BOC = m∠AOB = m∠POA = m∠DOE = 60º. Since all of the central angles (surrounding a point) must add to 360º, we know the m∠POE = 60º (360º  300º = 60º).
Since we have ΔDOE ΔDOE by SAS. By CPCTC, and we know that the last copy of the radii coincides with point P making the hexagon truly inscribed. Hexagon PABCDE has all vertices on circle O, has congruent interior angles (each equal 120º) and has all sides congruent. PABCDE is an inscribed regular hexagon by definition.
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