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TO inscribe a Regular Hexagon in a Circle.
To inscribe a Regular Hexagon in the Circle ABCDEF: draw the Diame∣ter AD, and putting the Foot of the Compass in the Point D, describe a Circle at the opening DG; which shall inter∣sect the Circle in the Points EC, then draw the Diameters EGB, CGF, and the Lines AB, AF, and the others.
Demonstration. It is evident that the
Triangles CDG, DGE, are equilate∣ral; wherefore the Angles CGD, DGE, and their opposites BGA, AGF, are each of them the third part of two Right; and that is 60 degrees. Now all the Angles which can be made about one Point is equal to four Right; that is to say, 360. So taking away four times 60, that is 240, from 360; there remains 120 degrees; for BGC, and FGE, whence they shall each be 60 degrees. So all the Angles at the Center being equal; all the Arks and all the Sides shall be equal; and each Angle A, B, C, &c. shall be composed of two Angles of Sixty; that is to say, One Hundred and Twenty degrees. They shall there∣fore be equal.
Coroll. The Side of a Hexagon is equal to the Semi diameter.
BEcause that the Side of an Hexagon is the Base of an Ark of Sixty degrees, and that is equal to the Semi-Diameter; its half is the Sine of Thirty; and it is with this Sine we begin the Tables of Sines. Euclid treateth of Hexagons in the last Book of his Elements.