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Compass and straightedge - the regular Heptadecagon

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Xuất bản 09/08/2016
One of the oldest and best-known topics in the history of mathematics are compass and straightedge (or compass and ruler) constructions. I will not illustrate them here, and neither will I explain why it's so remarkable that the regular heptadecagon, the 17 equal-sided polygon, actually CAN be constructed that way (you can find all you need just asking Saint Google about "compass and straightedge" and "regular heptadecagon"). Instead I prefer letting you get astonished by the amazing complexity of the procedure (which you can find described step by step here below). Enjoy! 1. Draw a line and choose on it points O and A. Draw a circle centered at O with radius OA, it will be the heptadecagon's circumcircle. 2. Draw the line through O that is perpendicular to OA and call B one of its two intersections with the circle. 3. Find the midpoint C of segment OB. 4. Find the midpoint P of segment OC. 5. Find the point V where segment PA cuts the circle centered at C with radius CO. 6. Draw the circle centered at P with radius AV locating F on PA and I on BO (there are two intersection points, choose the one nearer to O than to B). 7. Find H, the midpoint of arc FI. 8. Find J, the midpoint of arc HI, and K, symmetric to J with respect to P. 9. Locate G, the midpoint of the semicircle KJ (choose the one containing I). 10. Draw the line bisecting the right angle JPG, and let L denote its interception with OA. 11. Find the midpoint Z of segment AL. 12. Locate N where the circle centered at Z with radius ZL cuts segment CP. 13. Draw the circle centered at R, intersection point between PJ and OA, with radius RN. It strikes OA at point U, which is situated inside segment AZ very near to Z. 14. Draw the perpendicular line to OA through U. Call S its intersection point (choose one of the two possible ones) with the circle centered at O with radius OA. 15. A and S are vertices of the heptadecagon, use the arc AS to find the others. If you choose and keep a direction (clockwise for example) you should encounter exactly your starting point after three complete turns. 16. Draw the heptadecagon's sides. 17. If desired, erase construction lines.
math mathematics compass maths 17 construction ruler polygon straightedge regular heptadecagon