G.129 similar circle segments on equal chords are equal — Heath

III.24

Similar segments of circles on equal straight lines are equal to one another.

For let ( A E B ), ( C F D ) be similar segments of circles on equal straight lines / A B /, / C D /; I say that the segment ( A E B ) is equal to the segment ( C F D ).

For, if the segment ( A E B ) be applied to ( C F D ), and if the point A be placed on C and the straight line / A B / on / C D /, the point B will also coincide with the point D, because / A B / is equal to / C D /;

and, / A B / coinciding with / C D /,

the segment ( A E B ) will also coincide with ( C F D ).

A B E C D F G

For, if the straight line / A B / coincide with / C D / but the segment ( A E B ) do not coincide with ( C F D ), it will either fall within it, or outside it; or it will fall awry, as ( C G D ), and a circle cuts a circle at more points than two: which is impossible. [G.108]

Therefore, if the straight line / A B / be applied to / C D /, the segment ( A E B ) will not fail to coincide with ( C F D ) also;

therefore it will coincide with it and will be equal to it.
Therefore etc. Q. E. D.

Dependency Graph

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"G.6" [label="G.6", tooltip="G.6 (I.def.6): The extremities of a surface are lines or arcs.", fillcolor="#224422", URL="../G.6/index.html", target="_top"]; "G.7" [label="G.7", tooltip="G.7 (I.def.7): A plane surface lies evenly with the lines on itself.", fillcolor="#224422", URL="../G.7/index.html", target="_top"]; "G.8" [label="G.8", tooltip="G.8 (I.def.13): A boundary is the extremity of anything.", fillcolor="#224422", URL="../G.8/index.html", target="_top"]; "G.9" [label="G.9", tooltip="G.9 (I.def.14): A figure is that which is contained by any boundary or boundaries.", fillcolor="#224422", URL="../G.9/index.html", target="_top"]; "G.10" [label="G.10", tooltip="G.10 (I.def.15): A circle ( A B ) is a plane figure contained by one arc defined by a center A and a point B establishing its defining radius / A B /, such that all segments / A B /, / A C /, / A D...", fillcolor="#224422", URL="../G.10/index.html", target="_top"]; "G.11" [label="G.11", tooltip="G.11 (I.def.16): The point A within the circle ( A B ) from which all segments / A B /, / A C / to the arc are equal is called the center of the circle.", fillcolor="#224422", URL="../G.11/index.html", target="_top"]; "G.12" [label="G.12", tooltip="G.12 (I.def.19): Rectilineal figures are contained by segments: trilateral by three, quadrilateral by four, and multilateral by more than four.", fillcolor="#224422", URL="../G.12/index.html", target="_top"]; "G.13" [label="G.13", tooltip="G.13 (I.def.20): Of trilateral figures, an equilateral triangle < A B C > has three equal sides, an isosceles triangle < D E F > has two equal sides, and a scalene triangle < G H K > has three uneq...", fillcolor="#224422", URL="../G.13/index.html", target="_top"]; "G.14" [label="G.14", tooltip="G.14 (I.post.1): Construct a segment / A B / between any two points A, B.", fillcolor="#444422", URL="../G.14/index.html", target="_top"]; "G.15" [label="G.15", tooltip="G.15 (I.post.3): Construct a circle ( A B ) with center A and defining radius point B.", fillcolor="#444422", URL="../G.15/index.html", target="_top"]; "G.16" [label="G.16", tooltip="G.16 (I.cn.1): Things equal to the same thing are equal to one another.", fillcolor="#442222", URL="../G.16/index.html", target="_top"]; "G.17" [label="G.17", tooltip="G.17 (I.1): Construct an equilateral triangle on a given segment.", fillcolor="#222244", URL="../G.17/index.html", target="_top"]; "G.18" [label="G.18", tooltip="G.18 (I.post.2): Extend a segment / A B / continuously along a line [ A B ] to / B C /.", fillcolor="#444422", URL="../G.18/index.html", target="_top"]; "G.19" [label="G.19", tooltip="G.19 (I.cn.3): If equals are subtracted from equals, the remainders are equal.", fillcolor="#442222", URL="../G.19/index.html", target="_top"]; "G.20" [label="G.20", tooltip="G.20 (I.2): Construct at a given point a segment equal to a given segment.", fillcolor="#222244", URL="../G.20/index.html", target="_top"]; "G.21" [label="G.21", tooltip="G.21 (I.3): Cut off from the greater of two unequal segments a segment equal to the less.", fillcolor="#222244", URL="../G.21/index.html", target="_top"]; "G.22" [label="G.22", tooltip="G.22 (I.def.8): A plane angle is the inclination to one another of two lines or arcs in a plane that meet and do not lie on the same line.", fillcolor="#224422", URL="../G.22/index.html", target="_top"]; "G.23" [label="G.23", tooltip="G.23 (I.cn.4): Things / A B /, / D E / that coincide with one another are equal to one another.", fillcolor="#442222", URL="../G.23/index.html", target="_top"]; "G.24" [label="G.24", tooltip="G.24 (I.4): Triangles with two sides and the included angle equal are congruent (SAS).", fillcolor="#222244", URL="../G.24/index.html", target="_top"]; "G.25" [label="G.25", tooltip="G.25 (I.5): In an isosceles triangle, the angles at the base are equal, and if the equal sides are extended, the angles under the base are equal.", fillcolor="#222244", URL="../G.25/index.html", target="_top"]; "G.26" [label="G.26", tooltip="G.26 (I.cn.5): The whole is greater than the part.", fillcolor="#442222", URL="../G.26/index.html", target="_top"]; "G.28" [label="G.28", tooltip="G.28 (I.7): On the same segment and on the same side, there cannot be constructed two distinct triangles with both pairs of conterminous sides equal.", fillcolor="#222244", URL="../G.28/index.html", target="_top"]; "G.29" [label="G.29", tooltip="G.29 (I.8): Triangles with all three corresponding sides equal have their corresponding angles equal (SSS).", fillcolor="#222244", URL="../G.29/index.html", target="_top"]; "G.30" [label="G.30", tooltip="G.30 (I.def.9): A rectilineal angle is contained by two segments or lines.", fillcolor="#224422", URL="../G.30/index.html", target="_top"]; "G.31" [label="G.31", tooltip="G.31 (I.9): Bisect a given angle.", fillcolor="#222244", URL="../G.31/index.html", target="_top"]; "G.32" [label="G.32", tooltip="G.32 (I.10): Bisect a given segment.", fillcolor="#222244", URL="../G.32/index.html", target="_top"]; "G.33" [label="G.33", tooltip="G.33 (I.def.10): When a segment standing on a line makes the adjacent angles equal, each of the equal angles is a right angle, and the standing segment is perpendicular to the line.", fillcolor="#224422", URL="../G.33/index.html", target="_top"]; "G.34" [label="G.34", tooltip="G.34 (I.11): Construct a segment at right angles to a given line from a given point on it.", fillcolor="#222244", URL="../G.34/index.html", target="_top"]; "G.36" [label="G.36", tooltip="G.36 (I.cn.2): If equals are added to equals, the wholes are equal.", fillcolor="#442222", URL="../G.36/index.html", target="_top"]; "G.37" [label="G.37", tooltip="G.37 (I.13): A segment standing on a line makes either two right angles or adjacent angles summing to two right angles.", fillcolor="#222244", URL="../G.37/index.html", target="_top"]; "G.38" [label="G.38", tooltip="G.38 (I.post.4): All right angles are equal.", fillcolor="#444422", URL="../G.38/index.html", target="_top"]; "G.40" [label="G.40", tooltip="G.40 (I.15): Intersecting lines make their vertical angles equal.", fillcolor="#222244", URL="../G.40/index.html", target="_top"]; "G.41" [label="G.41", tooltip="G.41 (I.16): In any triangle, if one side is extended, the exterior angle is greater than either opposite interior angle.", fillcolor="#222244", URL="../G.41/index.html", target="_top"]; "G.98" [label="G.98", tooltip="G.98 (III.1): Find the center of a given circle.", fillcolor="#222244", URL="../G.98/index.html", target="_top"]; "G.102" [label="G.102", tooltip="G.102 (III.5): Two intersecting circles do not have the same center.", fillcolor="#222244", URL="../G.102/index.html", target="_top"]; "G.108" [label="G.108", tooltip="G.108 (III.10): A circle does not intersect another circle at more than two points.", fillcolor="#222244", URL="../G.108/index.html", target="_top"]; "G.126" [label="G.126", tooltip="G.126 (III.def.6): Asegment of a circle is thefigure contained by asegment and a arc of acircle.", fillcolor="#224422", URL="../G.126/index.html", target="_top"]; "G.127" [label="G.127", tooltip="G.127 (III.def.11): Similar segments of circles are those which admit equalangles, or in which the angles are equal.", fillcolor="#224422", URL="../G.127/index.html", target="_top"]; "G.128" [label="G.128", tooltip="G.128 (III.23): On the same segment and on the same side of it, there cannot be constructed two similar and unequal segments of circles.", fillcolor="#222244", URL="../G.128/index.html", target="_top"]; "G.129" [label="G.129", tooltip="G.129 (III.24): Similar segments of circles on equal segments are equal to one another.", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", URL="index.html", target="_top"]; "G.2" -> "G.1"; "G.3" -> "G.1"; "G.4" -> "G.1"; "G.10" -> "G.1"; "G.11" -> "G.1"; "G.14" -> "G.1"; "G.15" -> "G.1"; "G.17" -> "G.1"; "G.20" -> "G.1"; "G.21" -> "G.1"; "G.24" -> "G.1"; "G.25" -> "G.1"; "G.28" -> "G.1"; "G.29" -> "G.1"; "G.31" -> "G.1"; "G.32" -> "G.1"; "G.34" -> "G.1"; "G.37" -> "G.1"; "G.40" -> "G.1"; "G.108" -> "G.1"; "G.129" -> "G.1"; "G.3" -> "G.2"; "G.4" -> "G.2"; "G.6" -> "G.2"; "G.7" -> "G.2"; "G.18" -> "G.2"; "G.22" -> "G.2"; "G.30" -> "G.2"; "G.33" -> "G.2"; "G.34" -> "G.2"; "G.37" -> "G.2"; "G.40" -> "G.2"; "G.4" -> "G.3"; "G.6" -> "G.3"; "G.8" -> "G.3"; "G.20" -> "G.3"; "G.28" -> "G.3"; "G.29" -> "G.3"; "G.7" -> "G.4"; "G.10" -> "G.4"; "G.11" -> "G.4"; "G.12" -> "G.4"; "G.14" -> "G.4"; "G.17" -> "G.4"; "G.18" -> "G.4"; "G.20" -> "G.4"; "G.21" -> "G.4"; "G.22" -> "G.4"; "G.24" -> "G.4"; "G.25" -> "G.4"; "G.28" -> "G.4"; "G.29" -> "G.4"; "G.30" -> "G.4"; "G.31" -> "G.4"; "G.32" -> "G.4"; "G.33" -> "G.4"; "G.34" -> "G.4"; "G.37" -> "G.4"; "G.40" -> "G.4"; "G.41" -> "G.4"; "G.98" -> "G.4"; "G.126" -> "G.4"; "G.128" -> "G.4"; "G.129" -> "G.4"; "G.6" -> "G.5"; "G.7" -> "G.5"; "G.10" -> "G.6"; "G.10" -> "G.7"; "G.22" -> "G.7"; "G.9" -> "G.8"; "G.10" -> "G.9"; "G.12" -> "G.9"; "G.126" -> "G.9"; "G.11" -> "G.10"; "G.15" -> "G.10"; "G.17" -> "G.10"; "G.20" -> "G.10"; "G.21" -> "G.10"; "G.98" -> "G.10"; "G.102" -> "G.10"; "G.108" -> "G.10"; "G.126" -> "G.10"; "G.129" -> "G.10"; "G.15" -> "G.11"; "G.17" -> "G.11"; "G.20" -> "G.11"; "G.21" -> "G.11"; "G.98" -> "G.11"; "G.102" -> "G.11"; "G.108" -> "G.11"; "G.13" -> "G.12"; "G.17" -> "G.12"; "G.20" -> "G.12"; "G.24" -> "G.12"; "G.25" -> "G.12"; "G.28" -> "G.12"; "G.29" -> "G.12"; "G.31" -> "G.12"; "G.32" -> "G.12"; "G.34" -> "G.12"; "G.41" -> "G.12"; "G.17" -> "G.13"; "G.25" -> "G.13"; "G.17" -> "G.14"; "G.20" -> "G.14"; "G.25" -> "G.14"; "G.28" -> "G.14"; "G.31" -> "G.14"; "G.34" -> "G.14"; "G.41" -> "G.14"; "G.98" -> "G.14"; "G.102" -> "G.14"; "G.108" -> "G.14"; "G.128" -> "G.14"; "G.17" -> "G.15"; "G.20" -> "G.15"; "G.21" -> "G.15"; "G.17" -> "G.16"; "G.20" -> "G.16"; "G.21" -> "G.16"; "G.37" -> "G.16"; "G.40" -> "G.16"; "G.102" -> "G.16"; "G.20" -> "G.17"; "G.31" -> "G.17"; "G.32" -> "G.17"; "G.34" -> "G.17"; "G.20" -> "G.18"; "G.25" -> "G.18"; "G.41" -> "G.18"; "G.108" -> "G.18"; "G.20" -> "G.19"; "G.25" -> "G.19"; "G.40" -> "G.19"; "G.21" -> "G.20"; "G.25" -> "G.21"; "G.31" -> "G.21"; "G.34" -> "G.21"; "G.41" -> "G.21"; "G.24" -> "G.22"; "G.25" -> "G.22"; "G.28" -> "G.22"; "G.29" -> "G.22"; "G.30" -> "G.22"; "G.31" -> "G.22"; "G.32" -> "G.22"; "G.33" -> "G.22"; "G.34" -> "G.22"; "G.37" -> "G.22"; "G.40" -> "G.22"; "G.41" -> "G.22"; "G.127" -> "G.22"; "G.24" -> "G.23"; "G.29" -> "G.23"; "G.129" -> "G.23"; "G.25" -> "G.24"; "G.32" -> "G.24"; "G.41" -> "G.24"; "G.28" -> "G.25"; "G.28" -> "G.26"; "G.41" -> "G.26"; "G.98" -> "G.26"; "G.102" -> "G.26"; "G.29" -> "G.28"; "G.31" -> "G.29"; "G.34" -> "G.29"; "G.98" -> "G.29"; "G.31" -> "G.30"; "G.32" -> "G.31"; "G.41" -> "G.32"; "G.98" -> "G.32"; "G.108" -> "G.32"; "G.34" -> "G.33"; "G.37" -> "G.33"; "G.38" -> "G.33"; "G.40" -> "G.33"; "G.98" -> "G.33"; "G.37" -> "G.34"; "G.98" -> "G.34"; "G.108" -> "G.34"; "G.37" -> "G.36"; "G.40" -> "G.37"; "G.40" -> "G.38"; "G.41" -> "G.40"; "G.128" -> "G.41"; "G.108" -> "G.98"; "G.108" -> "G.102"; "G.129" -> "G.108"; "G.128" -> "G.126"; "G.128" -> "G.127"; "G.129" -> "G.127"; "G.129" -> "G.128"; }

Required for

G.133, G.134, G.135, G.136, G.137, G.163, G.164, G.167, G.168, G.246, G.551, G.552, G.556, G.557, G.558, G.559, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578, G.586, G.587, G.588, G.589, G.590, G.592, G.597, G.601