G.17 construct an equilateral triangle — Heath

I.1

On a given finite straight line to construct an equilateral triangle.
C D A B E

Let / A B / be the given finite straight line.

Thus it is required to construct an equilateral triangle on the straight line / A B /.

With centre A and distance / A B / let the circle ( B C D ) be described; [G.15]

again, with centre B and distance / B A / let the circle ( A C E ) be described; [G.15]

and from the point C, in which the circles cut one another, to the points A, B let the straight lines / C A /, / C B / be joined. [G.14]

Now, since the point A is the centre of the circle ( C D B ),

[G.10]/ A C / is equal to / A B /.

Again, since the point B is the centre of the circle ( C A E ),

[G.10]/ B C / is equal to / B A /.

But / C A / was also proved equal to / A B /; therefore each of the straight lines / C A /, / C B / is equal to / A B /.

And things which are equal to the same thing are also equal to one another; [G.16]

therefore / C A / is also equal to / C B /.

Therefore the three straight lines / C A /, / A B /, / B C / are equal to one another.

Therefore the triangle < A B C > is equilateral; and it has been constructed on the given finite straight line / A B /.

(Being) what it was required to do.

Dependency Graph

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Required for

G.20, G.21, G.25, G.27, G.28, G.29, G.31, G.32, G.34, G.35, 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.98, G.99, G.100, G.101, G.105, G.106, G.107, G.108, 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.129, G.130, G.133, G.134, G.135, G.136, G.137, 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.161, G.162, G.163, G.164, G.165, G.166, G.167, G.168, 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.585, G.586, G.587, G.588, G.589, G.590, G.592, G.594, G.595, G.597, G.600, G.601