PROPOSITION 14.

In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another.

Let ABDC be a circle, and let AB, CD be equal straight lines in it; I say that AB, CD are equally distant from the centre.

For let the centre of the circle ABDC be taken [III. 1], and let it be E; from E let EF, EG be drawn perpendicular to AB, CD, and let AE, EC be joined.

A B C D E F G

Then, since a straight line EF through the centre cuts a straight line AB not through the centre at right angles, it also bisects it. [III. 3]

Therefore AF is equal to FB; therefore AB is double of AF.

For the same reason

CD is also double of CG;

and AB is equal to CD;

therefore AF is also equal to CG.

And, since AE is equal to EC,

the square on AE is also equal to the square on EC.

But the squares on AF, EF are equal to the square on AE, for the angle at F is right; and the squares on EG, GC are equal to the square on EC, for the angle at G is right; [I. 47]

therefore the squares on AF, FE are equal to the squares on CG, GE,

of which the square on AF is equal to the square on CG, for AF is equal to CG;

therefore the square on FE which remains is equal to the square on EG,
therefore EF is equal to EG.

But in a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal; [III. Def. 4]

therefore AB, CD are equally distant from the centre.

Next, let the straight lines AB, CD be equally distant from the centre; that is, let EF be equal to EG.

I say that AB is also equal to CD.

For, with the same construction, we can prove, similarly, that AB is double of AF, and CD of CG.

And, since AE is equal to CE,

the square on AE is equal to the square on CE.

But the squares on EF, FA are equal to the square on AE, and the squares on EG, GC equal to the square on CE. [I. 47] Therefore the squares on EF, FA are equal to the squares on EG, GC, of which the square on EF is equal to the square on EG, for EF is equal to EG; therefore the square on AF which remains is equal to the square on CG;

therefore AF is equal to CG.

And AB is double of AF, and CD double of CG;

therefore AB is equal to CD.
Therefore etc. Q. E. D.

Dependency Graph

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"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.post.5" [label="I.post.5", fillcolor="#444422", tooltip="I.post.5: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet…", URL="../../../../volume_I/book_i/postulates/index.html#i-post-5", 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.14" [label="I.14", fillcolor="#222244", tooltip="I.14: If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be…", URL="../../../../volume_I/book_i/propositions/14/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.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.20" [label="I.20", fillcolor="#222244", tooltip="I.20: In any triangle two sides taken together in any manner are greater than the remaining one.", URL="../../../../volume_I/book_i/propositions/20/index.html", target="_top"]; "I.22" [label="I.22", fillcolor="#222244", tooltip="I.22: Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any…", URL="../../../../volume_I/book_i/propositions/22/index.html", target="_top"]; "I.23" [label="I.23", fillcolor="#222244", tooltip="I.23: On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.", URL="../../../../volume_I/book_i/propositions/23/index.html", target="_top"]; 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"I.34" [label="I.34", fillcolor="#222244", tooltip="I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.", URL="../../../../volume_I/book_i/propositions/34/index.html", target="_top"]; "I.35" [label="I.35", fillcolor="#222244", tooltip="I.35: Parallelograms which are on the same base and in the same parallels are equal to one another.", URL="../../../../volume_I/book_i/propositions/35/index.html", target="_top"]; "I.37" [label="I.37", fillcolor="#222244", tooltip="I.37: Triangles which are on the same base and in the same parallels are equal to one another.", URL="../../../../volume_I/book_i/propositions/37/index.html", target="_top"]; "I.41" [label="I.41", fillcolor="#222244", tooltip="I.41: If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.", URL="../../../../volume_I/book_i/propositions/41/index.html", target="_top"]; "I.46" [label="I.46", fillcolor="#222244", tooltip="I.46: On a given straight line to describe a square.", URL="../../../../volume_I/book_i/propositions/46/index.html", target="_top"]; "I.47" [label="I.47", fillcolor="#222244", tooltip="I.47: In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.", URL="../../../../volume_I/book_i/propositions/47/index.html", target="_top"]; "III.def.4" [label="III.def.4", fillcolor="#224422", tooltip="III.def.4: In a circle straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal.", URL="../../definitions/index.html#iii-def-4", target="_top"]; "III.1" [label="III.1", fillcolor="#222244", tooltip="III.1: To find the centre of a given circle.", URL="../01/index.html", 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="../03/index.html", target="_top"]; "III.14" [label="III.14", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="III.14: In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another.", URL="index.html", target="_top"]; 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"III.def.4" -> "I.def.10"; "III.1" -> "I.def.10"; "III.3" -> "I.def.10"; "I.def.21" -> "I.def.11"; "I.def.21" -> "I.def.12"; "I.def.14" -> "I.def.13"; "I.def.15" -> "I.def.14"; "I.4" -> "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.4" -> "I.def.15"; "III.1" -> "I.def.15"; "III.3" -> "I.def.15"; "III.14" -> "I.def.15"; "I.def.17" -> "I.def.16"; "I.post.3" -> "I.def.16"; "III.def.4" -> "I.def.16"; "III.1" -> "I.def.16"; "III.3" -> "I.def.16"; "I.34" -> "I.def.17"; "I.def.20" -> "I.def.19"; "I.def.21" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.4" -> "I.def.19"; "I.8" -> "I.def.19"; "I.16" -> "I.def.19"; "I.18" -> "I.def.19"; "I.19" -> "I.def.19"; "I.20" -> "I.def.19"; "I.22" -> "I.def.19"; "I.26" -> "I.def.19"; "I.37" -> "I.def.19"; "I.41" -> "I.def.19"; "I.def.22" -> "I.def.20"; "I.1" -> "I.def.20"; "I.5" -> "I.def.20"; "I.47" -> "I.def.21"; "I.35" -> "I.def.22"; "I.46" -> "I.def.22"; "I.47" -> "I.def.22"; "I.27" -> "I.def.23"; "I.29" -> "I.def.23"; "I.31" -> "I.def.23"; "I.35" -> "I.def.23"; "I.37" -> "I.def.23"; "I.41" -> "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.1" -> "I.post.3"; "I.2" -> "I.post.3"; "I.3" -> "I.post.3"; "I.14" -> "I.post.4"; "I.15" -> "I.post.4"; "I.29" -> "I.post.5"; "I.1" -> "I.cn.1"; "I.2" -> "I.cn.1"; "I.3" -> "I.cn.1"; "I.13" -> "I.cn.1"; "I.14" -> "I.cn.1"; "I.15" -> "I.cn.1"; "I.29" -> "I.cn.1"; "I.35" -> "I.cn.1"; "I.13" -> "I.cn.2"; "I.29" -> "I.cn.2"; "I.34" -> "I.cn.2"; "I.35" -> "I.cn.2"; "I.47" -> "I.cn.2"; "I.2" -> "I.cn.3"; "I.14" -> "I.cn.3"; "I.15" -> "I.cn.3"; "I.35" -> "I.cn.3"; "I.4" -> "I.cn.4"; "I.16" -> "I.cn.5"; "I.20" -> "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.22" -> "I.3"; "I.5" -> "I.4"; "I.10" -> "I.4"; "I.16" -> "I.4"; "I.26" -> "I.4"; "I.34" -> "I.4"; "I.35" -> "I.4"; "I.47" -> "I.4"; "I.7" -> "I.5"; "I.19" -> "I.5"; "I.20" -> "I.5"; "III.3" -> "I.5"; "I.8" -> "I.7"; "I.9" -> "I.8"; "I.11" -> "I.8"; "I.23" -> "I.8"; "III.1" -> "I.8"; "III.3" -> "I.8"; "I.10" -> "I.9"; "I.16" -> "I.10"; "I.13" -> "I.11"; "I.46" -> "I.11"; "I.14" -> "I.13"; "I.15" -> "I.13"; "I.29" -> "I.13"; "I.47" -> "I.14"; "I.16" -> "I.15"; "I.29" -> "I.15"; "I.18" -> "I.16"; "I.26" -> "I.16"; "I.27" -> "I.16"; "I.19" -> "I.18"; "I.20" -> "I.19"; "I.22" -> "I.20"; "I.23" -> "I.22"; "I.31" -> "I.23"; "I.34" -> "I.26"; "III.3" -> "I.26"; "I.31" -> "I.27"; "I.34" -> "I.29"; "I.35" -> "I.29"; "I.46" -> "I.29"; "I.37" -> "I.31"; "I.46" -> "I.31"; "I.35" -> "I.34"; "I.37" -> "I.34"; "I.41" -> "I.34"; "I.46" -> "I.34"; "I.37" -> "I.35"; "I.41" -> "I.37"; "I.47" -> "I.41"; "I.47" -> "I.46"; "III.14" -> "I.47"; "III.14" -> "III.def.4"; "III.14" -> "III.1"; "III.14" -> "III.3"; }

Required for

III.15

See Commentary on Proposition III.14.