Being one of integral parts of the school program, geometrical tasks on creation of regular polygons are rather trivial. As a rule, construction is conducted by description of a polygon in a circle which is drawn by the first. But what to do if the circle is set, and the figure is very difficult?
It is required to you
- - ruler;
- - compasses;
- - pencil;
- - sheet of paper.
Instruction
1. To the available circle construct a chord. Draw any piece of a straight line so that it had two points of intersection with a circle. Define these points as A and B.
2. Construct a piece of direct, perpendicular AB and dividing it in a point of intersection into two equal parts. Put a compasses needle in A point. Put a leg with a slate pencil in B point, or in any point of a piece which is closer to B than to A. Draw a circle. Without changing solution of legs of compasses establish its needle in B point. Draw one more circle. The drawn circles will be crossed in two points. Carry out through them a straight line piece. Designate a point of intersection of this piece with AB piece as C. Designate points of intersection of this piece with an initial circle as D and E.
3. Construct the perpendicular to DE piece halving it. Make the actions similar to that were described in the previous step, in relation to DE piece. Let the drawn piece cross DE in O point. This point will be the center of a circle. Also designate points of intersection of the constructed perpendicular with an initial circle as F and G.
4. Establish solution of legs of compasses so that the distance between their ends was equal to the radius of an initial circle. For this purpose place a compasses needle in one of points of A, B, D, E, F or G. Place the end of a leg with a slate pencil in O point.
5. Construct the correct hexagon. Establish a compasses needle in any point of the line of a circle. Designate this point of H. In the direction of the movement clockwise make compasses an arc-shaped notch so that it crossed the line of a circle. Designate this point of I. Move a compasses needle to a point of I. Again make a notch on a circle and designate the received J point. The same way construct points of K, L, M. Consistently in pairs connect points of H, I, J, K, L, M, H. The received figure is the correct hexagon entered in the set circle.
6. Find missing points of tops of corners of a dodecagon. To pieces of HI, IJ, JK construct the perpendiculars halving them so that the constructed pieces crossed O circle in two points. Designate the received points by the letters N, O, P, Q, R, S, starting with that that is behind H point on a circle in the direction of the movement of an hour hand.
7. Construct the correct dodecagon, inscribed in a circle. In pairs connect points of H, N, I, O, J, P, K, Q, L, R, M, S, H pieces. The polygon of HNIOJPKQLRMS is a required dodecagon.