G.34 construct a perpendicular at a point on a line — Heath

I.11

To draw a straight line at right angles to a given straight line from a given point on it.
F A D C E B

Let / A B / be the given straight line, and C the given point on it.

Thus it is required to draw from the point C a straight line at right angles to the straight line / A B /.

Let a point D be taken at random on / A C /; let / C E / be made equal to / C D /; [G.21]

on / D E / let the equilateral triangle < F D E > be constructed, [G.17]

and let / F C / be joined; [G.14]

I say that the straight line / F C / has been drawn at right angles to the given straight line / A B / from C the given point on it.

For, since / D C / is equal to / C E /, and / C F / is common,

the two sides / D C /, / C F / are equal to the two sides / E C /, / C F / respectively;

and the base / D F / is equal to the base / F E /;

[G.29]therefore the angle < D C F ) is equal to the angle < E C F );

and they are adjacent angles.

But, 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; [G.33]

therefore each of the angles < D C F ), < F C E ) is right.

Therefore the straight line / C F / has been drawn at right angles to the given straight line / A B / from the given point C on it.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "G.1" [label="G.1", tooltip="G.1 (I.def.1): A point A is that which has no part.", fillcolor="#224422", URL="../G.1/index.html", target="_top"]; "G.2" [label="G.2", tooltip="G.2 (I.def.2): A line [ A B ] is a projection of a relationship of two points A, B.", fillcolor="#224422", URL="../G.2/index.html", target="_top"]; "G.3" [label="G.3", tooltip="G.3 (I.def.3): The extremities of a segment / A B / are points A, B.", fillcolor="#224422", URL="../G.3/index.html", target="_top"]; "G.4" [label="G.4", tooltip="G.4 (I.def.4): A segment / A B / is a portion of a line [ A B ] between two points A, B on the line.", fillcolor="#224422", URL="../G.4/index.html", target="_top"]; "G.5" [label="G.5", tooltip="G.5 (I.def.5): A surface is that which has length and breadth only.", fillcolor="#224422", URL="../G.5/index.html", target="_top"]; "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.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.", 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.34" -> "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.33" -> "G.2"; "G.34" -> "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.33" -> "G.4"; "G.34" -> "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.11" -> "G.10"; "G.15" -> "G.10"; "G.17" -> "G.10"; "G.20" -> "G.10"; "G.21" -> "G.10"; "G.15" -> "G.11"; "G.17" -> "G.11"; "G.20" -> "G.11"; "G.21" -> "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.34" -> "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.34" -> "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.20" -> "G.17"; "G.34" -> "G.17"; "G.20" -> "G.18"; "G.25" -> "G.18"; "G.20" -> "G.19"; "G.25" -> "G.19"; "G.21" -> "G.20"; "G.25" -> "G.21"; "G.34" -> "G.21"; "G.24" -> "G.22"; "G.25" -> "G.22"; "G.28" -> "G.22"; "G.29" -> "G.22"; "G.33" -> "G.22"; "G.34" -> "G.22"; "G.24" -> "G.23"; "G.29" -> "G.23"; "G.25" -> "G.24"; "G.28" -> "G.25"; "G.28" -> "G.26"; "G.29" -> "G.28"; "G.34" -> "G.29"; "G.34" -> "G.33"; }

Required for

G.37, G.39, G.40, G.41, G.42, G.43, G.44, G.45, G.46, G.47, G.48, G.49, G.50, G.51, G.53, G.54, G.56, G.57, G.58, G.59, G.60, G.62, G.64, G.65, G.66, G.67, G.68, G.69, G.70, G.71, G.72, G.73, G.74, G.75, G.79, G.80, G.82, G.83, G.84, G.85, G.87, G.88, G.89, G.90, G.91, G.92, G.93, G.94, G.95, G.97, G.99, G.100, G.101, G.105, G.106, G.110, G.112, G.114, G.116, G.118, G.119, G.120, G.121, G.123, G.124, G.125, G.128, G.134, G.136, G.139, G.140, G.141, G.142, G.143, G.144, G.145, G.149, G.151, G.154, G.157, G.158, G.159, G.160, G.162, G.163, G.164, G.165, G.167, G.210, G.211, G.212, G.213, G.214, G.215, G.216, G.218, G.219, G.220, G.221, G.222, G.223, G.225, G.226, G.227, G.228, G.229, G.231, G.232, G.234, G.235, G.236, G.237, G.238, G.239, G.240, G.241, G.243, G.244, G.245, G.246, G.345, G.377, G.378, G.381, G.382, G.383, G.384, G.385, G.386, G.389, G.390, G.393, G.394, G.395, G.396, G.397, G.398, G.399, G.400, G.401, G.402, G.403, G.404, 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.429, 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.455, 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.501, G.506, G.507, G.508, G.510, G.511, G.512, G.513, G.514, G.517, G.518, G.520, G.522, G.525, G.526, G.527, G.528, G.530, G.531, G.532, G.536, G.538, G.539, G.540, G.541, G.542, G.543, G.544, G.545, G.546, G.547, G.549, G.550, G.551, G.552, G.554, G.555, G.556, G.557, G.558, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.574, G.577, G.578, G.579, G.580, G.582, G.583, G.584, G.586, G.587, G.588, G.589, G.590, G.592, G.594, G.595, G.597, G.600, G.601