G.147 fit a chord in a circle equal to a segment — Heath

IV.1

Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.

Let ( A B C ) be the given circle, and D the given straight line not greater than the diameter of the circle; thus it is required to fit into the circle ( A B C ) a straight line equal to the straight line D.

A B C E F D

Let a diameter / B C / of the circle ( A B C ) be drawn.

Then, if / B C / is equal to D, that which was enjoined will have been done; for / B C / has been fitted into the circle ( A B C ) equal to the straight line D.

But, if / B C / is greater than D, let / C E / be made equal to D, and with centre C and distance / C E / let the circle ( E A F ) be described; let / C A / be joined.

Then, since the point C is the centre of the circle ( E A F ),

/ C A / is equal to / C E /.

But / C E / is equal to D;

therefore D is also equal to / C A /.

Therefore into the given circle ( A B C ) there has been fitted / C A / equal to the given straight line 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.61" [label="G.61", tooltip="G.61 (I.def.17): A diameter of a circle is any segment drawn through the center and terminated in both directions by the arc of the circle; it bisects the circle.", fillcolor="#224422", URL="../G.61/index.html", target="_top"]; "G.146" [label="G.146", tooltip="G.146 (IV.def.7): Asegment is said to befitted into a circle when itsextremities are on the arc of thecircle.", fillcolor="#224422", URL="../G.146/index.html", target="_top"]; "G.147" [label="G.147", tooltip="G.147 (IV.1): Fit into a given circle a chord equal to a given segment not greater than the diameter of the circle.", 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.3" -> "G.2"; "G.4" -> "G.2"; "G.6" -> "G.2"; "G.7" -> "G.2"; "G.18" -> "G.2"; "G.4" -> "G.3"; "G.6" -> "G.3"; "G.8" -> "G.3"; "G.20" -> "G.3"; "G.146" -> "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.61" -> "G.4"; "G.146" -> "G.4"; "G.147" -> "G.4"; "G.6" -> "G.5"; "G.7" -> "G.5"; "G.10" -> "G.6"; "G.146" -> "G.6"; "G.10" -> "G.7"; "G.9" -> "G.8"; "G.10" -> "G.9"; "G.12" -> "G.9"; "G.11" -> "G.10"; "G.15" -> "G.10"; "G.17" -> "G.10"; "G.20" -> "G.10"; "G.21" -> "G.10"; "G.61" -> "G.10"; "G.146" -> "G.10"; "G.147" -> "G.10"; "G.15" -> "G.11"; "G.17" -> "G.11"; "G.20" -> "G.11"; "G.21" -> "G.11"; "G.61" -> "G.11"; "G.13" -> "G.12"; "G.17" -> "G.12"; "G.20" -> "G.12"; "G.17" -> "G.13"; "G.17" -> "G.14"; "G.20" -> "G.14"; "G.147" -> "G.14"; "G.17" -> "G.15"; "G.20" -> "G.15"; "G.21" -> "G.15"; "G.147" -> "G.15"; "G.17" -> "G.16"; "G.20" -> "G.16"; "G.21" -> "G.16"; "G.147" -> "G.16"; "G.20" -> "G.17"; "G.20" -> "G.18"; "G.20" -> "G.19"; "G.21" -> "G.20"; "G.147" -> "G.21"; "G.147" -> "G.61"; "G.147" -> "G.146"; }

Required for

G.162, G.163, G.164, G.385, G.390, G.396, G.397, G.398, G.399, G.400, G.401, G.402, G.405, G.406, G.407, G.408, G.409, G.410, G.411, G.412, G.413, G.414, G.415, G.416, G.417, G.418, G.419, G.420, G.421, G.423, G.425, G.427, G.431, G.433, G.434, G.435, G.436, G.437, G.438, G.439, G.440, G.441, G.442, G.443, G.444, G.445, G.446, G.447, G.448, G.449, G.450, G.451, G.452, G.453, G.457, G.458, G.459, G.460, G.461, G.462, G.463, G.464, G.465, G.466, G.467, G.468, G.470, G.472, G.474, G.476, G.477, G.478, G.479, G.480, G.481, G.482, G.483, G.484, G.485, G.486, G.487, G.488, G.489, G.490, G.491, G.492, G.493, G.494, G.495, G.496, G.497, G.498, G.499, G.500, G.528, G.584, G.589, G.597, G.600, G.601