PROPOSITION 16.

Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.

Let ABCD, EFGH be the two given circles about the same centre K ; thus it is required to inscribe in the greater circle ABCD an equilateral polygon with an even number of sides which does not touch the circle EFGH.

H A L B E K G M D N C F

For let the straight line BKD be drawn through the centre K, and from the point G let GA be drawn at right angles to the straight line BD and carried through to C ;

[III. 16, Por.]therefore AC touches the circle EFGH.

Then, bisecting the circumference BAD, bisecting the half of it, and doing this continually, we shall leave a circumference less than AD. [X. 1] Let such be left, and let it be LD ; from L let LM be drawn perpendicular to BD and carried through to N, and let LD, DN be joined ;

[III. 3, I. 4]therefore LD is equal to DN.
Now, since LN is parallel to AC, and AC touches the circle EFGH,
therefore LN does not touch the circle EFGH ; therefore LD, DN are far from touching the circle EFGH.
If then we fit into the circle ABCD straight lines equal to the straight line LD and placed continuously, there will be inscribed in the circle ABCD an equilateral polygon with an even number of sides which does not touch the lesser circle EFGH. Q. E. F.

Dependency Graph

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"I.def.5" [label="I.def.5", fillcolor="#224422", tooltip="I.def.5: A surface is that which has length and breadth only.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-5", target="_top"]; "I.def.6" [label="I.def.6", fillcolor="#224422", tooltip="I.def.6: The extremities of a surface are lines.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-6", target="_top"]; "I.def.7" [label="I.def.7", fillcolor="#224422", tooltip="I.def.7: A plane surface is a surface which lies evenly with the straight lines on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-7", target="_top"]; "I.def.8" [label="I.def.8", fillcolor="#224422", tooltip="I.def.8: A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-8", target="_top"]; "I.def.9" [label="I.def.9", fillcolor="#224422", tooltip="I.def.9: And when the lines containing the angle are straight, the angle is called rectilineal.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-9", target="_top"]; "I.def.10" [label="I.def.10", fillcolor="#224422", tooltip="I.def.10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-10", target="_top"]; "I.def.12" [label="I.def.12", fillcolor="#224422", tooltip="I.def.12: An acute angle is an angle less than a right angle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-12", target="_top"]; "I.def.13" [label="I.def.13", fillcolor="#224422", tooltip="I.def.13: A boundary is that which is an extremity of anything.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-13", target="_top"]; "I.def.14" [label="I.def.14", fillcolor="#224422", tooltip="I.def.14: A figure is that which is contained by any boundary or boundaries.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-14", target="_top"]; "I.def.15" [label="I.def.15", fillcolor="#224422", tooltip="I.def.15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-15", target="_top"]; "I.def.16" [label="I.def.16", fillcolor="#224422", tooltip="I.def.16: And the point is called the centre of the circle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-16", target="_top"]; "I.def.17" [label="I.def.17", fillcolor="#224422", tooltip="I.def.17: A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-17", target="_top"]; "I.def.19" [label="I.def.19", fillcolor="#224422", tooltip="I.def.19: Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-19", target="_top"]; "I.def.20" [label="I.def.20", fillcolor="#224422", tooltip="I.def.20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-20", target="_top"]; "I.def.23" [label="I.def.23", fillcolor="#224422", tooltip="I.def.23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-23", target="_top"]; "I.post.1" [label="I.post.1", fillcolor="#444422", tooltip="I.post.1: To draw a straight line from any point to any point.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-1", target="_top"]; "I.post.2" [label="I.post.2", fillcolor="#444422", tooltip="I.post.2: To produce a finite straight line continuously in a straight line.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-2", target="_top"]; "I.post.3" [label="I.post.3", fillcolor="#444422", tooltip="I.post.3: To describe a circle with any centre and distance.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-3", target="_top"]; "I.post.4" [label="I.post.4", fillcolor="#444422", tooltip="I.post.4: That all right angles are equal to one another.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-4", target="_top"]; "I.cn.1" [label="I.cn.1", fillcolor="#442222", tooltip="I.cn.1: Things which are equal to the same thing are also equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-1", target="_top"]; "I.cn.2" [label="I.cn.2", fillcolor="#442222", tooltip="I.cn.2: If equals be added to equals, the wholes are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-2", target="_top"]; "I.cn.3" [label="I.cn.3", fillcolor="#442222", tooltip="I.cn.3: If equals be subtracted from equals, the remainders are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-3", target="_top"]; "I.cn.4" [label="I.cn.4", fillcolor="#442222", tooltip="I.cn.4: Things which coincide with one another are equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-4", target="_top"]; "I.cn.5" [label="I.cn.5", fillcolor="#442222", tooltip="I.cn.5: The whole is greater than the part.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-5", target="_top"]; "I.1" [label="I.1", fillcolor="#222244", tooltip="I.1: On a given finite straight line to construct an equilateral triangle.", URL="../../../../volume_I/book_i/propositions/01/index.html", target="_top"]; "I.2" [label="I.2", fillcolor="#222244", tooltip="I.2: To place at a given point (as an extremity) a straight line equal to a given straight line.", URL="../../../../volume_I/book_i/propositions/02/index.html", target="_top"]; "I.3" [label="I.3", fillcolor="#222244", tooltip="I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.", URL="../../../../volume_I/book_i/propositions/03/index.html", target="_top"]; "I.4" [label="I.4", fillcolor="#222244", tooltip="I.4: If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the…", URL="../../../../volume_I/book_i/propositions/04/index.html", target="_top"]; "I.5" [label="I.5", fillcolor="#222244", tooltip="I.5: In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one…", URL="../../../../volume_I/book_i/propositions/05/index.html", target="_top"]; "I.7" [label="I.7", fillcolor="#222244", tooltip="I.7: Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its…", URL="../../../../volume_I/book_i/propositions/07/index.html", target="_top"]; "I.8" [label="I.8", fillcolor="#222244", tooltip="I.8: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the…", URL="../../../../volume_I/book_i/propositions/08/index.html", target="_top"]; "I.9" [label="I.9", fillcolor="#222244", tooltip="I.9: To bisect a given rectilineal angle.", URL="../../../../volume_I/book_i/propositions/09/index.html", target="_top"]; "I.10" [label="I.10", fillcolor="#222244", tooltip="I.10: To bisect a given finite straight line.", URL="../../../../volume_I/book_i/propositions/10/index.html", target="_top"]; "I.11" [label="I.11", fillcolor="#222244", tooltip="I.11: To draw a straight line at right angles to a given straight line from a given point on it.", URL="../../../../volume_I/book_i/propositions/11/index.html", target="_top"]; "I.13" [label="I.13", fillcolor="#222244", tooltip="I.13: If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.", URL="../../../../volume_I/book_i/propositions/13/index.html", target="_top"]; "I.15" [label="I.15", fillcolor="#222244", tooltip="I.15: If two straight lines cut one another, they make the vertical angles equal to one another.", URL="../../../../volume_I/book_i/propositions/15/index.html", target="_top"]; "I.16" [label="I.16", fillcolor="#222244", tooltip="I.16: In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.", URL="../../../../volume_I/book_i/propositions/16/index.html", target="_top"]; "I.17" [label="I.17", fillcolor="#222244", tooltip="I.17: In any triangle two angles taken together in any manner are less than two right angles.", URL="../../../../volume_I/book_i/propositions/17/index.html", target="_top"]; "I.18" [label="I.18", fillcolor="#222244", tooltip="I.18: In any triangle the greater side subtends the greater angle.", URL="../../../../volume_I/book_i/propositions/18/index.html", target="_top"]; "I.19" [label="I.19", fillcolor="#222244", tooltip="I.19: In any triangle the greater angle is subtended by the greater side.", URL="../../../../volume_I/book_i/propositions/19/index.html", target="_top"]; "I.26" [label="I.26", fillcolor="#222244", tooltip="I.26: If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one…", URL="../../../../volume_I/book_i/propositions/26/index.html", target="_top"]; "III.def.2" [label="III.def.2", fillcolor="#224422", tooltip="III.def.2: A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-2", target="_top"]; "III.def.7" [label="III.def.7", fillcolor="#224422", tooltip="III.def.7: An angle of a segment is that contained by a straight line and a circumference of a circle.", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-7", target="_top"]; "III.3" [label="III.3", fillcolor="#222244", tooltip="III.3: If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at right angles; and if it cut it at right angles, it also bisects…", URL="../../../../volume_II/book_iii/propositions/03/index.html", target="_top"]; "III.16" [label="III.16", fillcolor="#222244", tooltip="III.16: The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the…", URL="../../../../volume_II/book_iii/propositions/16/index.html", target="_top"]; "IV.def.3" [label="IV.def.3", fillcolor="#224422", tooltip="IV.def.3: A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-3", target="_top"]; "IV.def.7" [label="IV.def.7", fillcolor="#224422", tooltip="IV.def.7: A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-7", target="_top"]; "V.def.1" [label="V.def.1", fillcolor="#224422", tooltip="V.def.1: A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-1", target="_top"]; "V.def.4" [label="V.def.4", fillcolor="#224422", tooltip="V.def.4: Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-4", target="_top"]; "VII.def.1" [label="VII.def.1", fillcolor="#224422", tooltip="VII.def.1: An unit is that by virtue of which each of the things that exist is called one.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-1", target="_top"]; "VII.def.2" [label="VII.def.2", fillcolor="#224422", tooltip="VII.def.2: A number is a multitude composed of units.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-2", target="_top"]; "VII.def.5" [label="VII.def.5", fillcolor="#224422", tooltip="VII.def.5: The greater number is a multiple of the less when it is measured by the less.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-5", target="_top"]; "VII.def.6" [label="VII.def.6", fillcolor="#224422", tooltip="VII.def.6: An even number is that which is divisible into two equal parts.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-6", target="_top"]; "VII.def.15" [label="VII.def.15", fillcolor="#224422", tooltip="VII.def.15: A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-15", target="_top"]; "X.1" [label="X.1", fillcolor="#222244", tooltip="X.1: Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and…", URL="../../../book_x/propositions/001/index.html", target="_top"]; "XII.16" [label="XII.16", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XII.16: Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.", URL="index.html", target="_top"]; "I.def.3" -> "I.def.1"; "I.def.4" -> "I.def.1"; "I.def.15" -> "I.def.1"; "I.def.16" -> "I.def.1"; "I.post.1" -> "I.def.1"; "I.2" -> "I.def.1"; "I.7" -> "I.def.1"; "I.11" -> "I.def.1"; "I.def.3" -> "I.def.2"; "I.def.4" -> "I.def.2"; "I.def.6" -> "I.def.2"; "I.def.8" -> "I.def.2"; "I.def.9" -> "I.def.2"; "I.def.4" -> "I.def.3"; "I.def.13" -> "I.def.3"; "I.2" -> "I.def.3"; "I.7" -> "I.def.3"; "III.16" -> "I.def.3"; "IV.def.7" -> "I.def.3"; "I.def.7" -> "I.def.4"; "I.def.8" -> "I.def.4"; "I.def.10" -> "I.def.4"; "I.def.15" -> "I.def.4"; "I.def.17" -> "I.def.4"; "I.def.19" -> "I.def.4"; "I.def.23" -> "I.def.4"; "I.post.1" -> "I.def.4"; "I.post.2" -> "I.def.4"; "I.1" -> "I.def.4"; "I.2" -> "I.def.4"; "I.3" -> "I.def.4"; "I.4" -> "I.def.4"; "I.5" -> "I.def.4"; "I.7" -> "I.def.4"; "I.8" -> "I.def.4"; "I.10" -> "I.def.4"; "I.11" -> "I.def.4"; "I.13" -> "I.def.4"; "I.15" -> "I.def.4"; "III.def.2" -> "I.def.4"; "III.def.7" -> "I.def.4"; "III.3" -> "I.def.4"; "III.16" -> "I.def.4"; "IV.def.7" -> "I.def.4"; "I.def.6" -> "I.def.5"; "I.def.7" -> "I.def.5"; "I.def.15" -> "I.def.6"; "I.def.8" -> "I.def.7"; "I.def.15" -> "I.def.7"; "I.def.23" -> "I.def.7"; "I.def.9" -> "I.def.8"; "I.def.10" -> "I.def.8"; "I.def.12" -> "I.def.8"; "I.4" -> "I.def.8"; "I.5" -> "I.def.8"; "I.8" -> "I.def.8"; "I.13" -> "I.def.8"; "I.15" -> "I.def.8"; "I.16" -> "I.def.8"; "I.17" -> "I.def.8"; "I.18" -> "I.def.8"; "I.19" -> "I.def.8"; "I.26" -> "I.def.8"; "III.16" -> "I.def.8"; "IV.def.3" -> "I.def.8"; "I.9" -> "I.def.9"; "III.16" -> "I.def.9"; "I.def.12" -> "I.def.10"; "I.post.4" -> "I.def.10"; "I.11" -> "I.def.10"; "I.13" -> "I.def.10"; "I.17" -> "I.def.10"; "III.3" -> "I.def.10"; "III.16" -> "I.def.10"; "III.16" -> "I.def.12"; "I.def.14" -> "I.def.13"; "I.def.15" -> "I.def.14"; "I.4" -> "I.def.14"; "IV.def.3" -> "I.def.14"; "I.def.16" -> "I.def.15"; "I.def.17" -> "I.def.15"; "I.post.3" -> "I.def.15"; "I.1" -> "I.def.15"; "I.3" -> "I.def.15"; "III.def.2" -> "I.def.15"; "III.def.7" -> "I.def.15"; "III.3" -> "I.def.15"; "III.16" -> "I.def.15"; "IV.def.3" -> "I.def.15"; "IV.def.7" -> "I.def.15"; "XII.16" -> "I.def.15"; "I.def.17" -> "I.def.16"; "I.post.3" -> "I.def.16"; "III.3" -> "I.def.16"; "XII.16" -> "I.def.16"; "III.16" -> "I.def.17"; "I.def.20" -> "I.def.19"; "I.4" -> "I.def.19"; "I.8" -> "I.def.19"; "I.16" -> "I.def.19"; "I.17" -> "I.def.19"; "I.18" -> "I.def.19"; "I.19" -> "I.def.19"; "I.26" -> "I.def.19"; "IV.def.3" -> "I.def.19"; "I.1" -> "I.def.20"; "I.5" -> "I.def.20"; "XII.16" -> "I.def.20"; "XII.16" -> "I.def.23"; "I.1" -> "I.post.1"; "I.2" -> "I.post.1"; "I.5" -> "I.post.1"; "I.7" -> "I.post.1"; "I.16" -> "I.post.1"; "I.2" -> "I.post.2"; "I.5" -> "I.post.2"; "I.16" -> "I.post.2"; "I.17" -> "I.post.2"; "I.1" -> "I.post.3"; "I.2" -> "I.post.3"; "I.3" -> "I.post.3"; "I.15" -> "I.post.4"; "I.1" -> "I.cn.1"; "I.2" -> "I.cn.1"; "I.3" -> "I.cn.1"; "I.13" -> "I.cn.1"; "I.15" -> "I.cn.1"; "I.13" -> "I.cn.2"; "I.2" -> "I.cn.3"; "I.15" -> "I.cn.3"; "I.4" -> "I.cn.4"; "I.16" -> "I.cn.5"; "I.2" -> "I.1"; "I.10" -> "I.1"; "I.11" -> "I.1"; "I.3" -> "I.2"; "I.5" -> "I.3"; "I.9" -> "I.3"; "I.11" -> "I.3"; "I.16" -> "I.3"; "I.18" -> "I.3"; "I.5" -> "I.4"; "I.10" -> "I.4"; "I.16" -> "I.4"; "I.26" -> "I.4"; "XII.16" -> "I.4"; "I.7" -> "I.5"; "I.19" -> "I.5"; "III.3" -> "I.5"; "III.16" -> "I.5"; "I.8" -> "I.7"; "I.9" -> "I.8"; "I.11" -> "I.8"; "III.3" -> "I.8"; "I.10" -> "I.9"; "I.16" -> "I.10"; "I.13" -> "I.11"; "I.15" -> "I.13"; "I.17" -> "I.13"; "I.16" -> "I.15"; "I.18" -> "I.16"; "I.26" -> "I.16"; "III.16" -> "I.17"; "I.19" -> "I.18"; "III.16" -> "I.19"; "III.3" -> "I.26"; "III.16" -> "III.def.2"; "III.16" -> "III.def.7"; "XII.16" -> "III.3"; "XII.16" -> "III.16"; "XII.16" -> "IV.def.3"; "XII.16" -> "IV.def.7"; "VII.def.5" -> "V.def.1"; "X.1" -> "V.def.4"; "VII.def.2" -> "VII.def.1"; "VII.def.15" -> "VII.def.1"; "VII.def.5" -> "VII.def.2"; "VII.def.15" -> "VII.def.2"; "X.1" -> "VII.def.5"; "X.1" -> "VII.def.6"; "XII.16" -> "VII.def.6"; "X.1" -> "VII.def.15"; "XII.16" -> "X.1"; }

See Commentary on Proposition XII.16.