G.504 planes cut common — Heath

XI.3

If two planes cut one another, their common section is a straight line.

For let the two planes < A B >, < B C > cut one another, and let the line / D B / be their common section; I say that the line / D B / is a straight line.

B F E D A C

For, if not, from D to B let the straight line / D E B / be joined in the plane < A B >, and in the plane < B C > the straight line / D F B /.

Then the two straight lines / D E B /, / D F B / will have the same extremities, and will clearly enclose an area:
which is absurd.
Therefore / D E B /, / D F B / are not straight lines.
Similarly we can prove that neither will there be any other straight line joined from D to B except / D B / the common section of the planes < A B >, < B C >.
Therefore etc. Q. E. D.

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.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.504" [label="G.504", tooltip="G.504 (XI.3): If two planes cut one another, their common section is a segment.", 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.3" -> "G.2"; "G.4" -> "G.2"; "G.7" -> "G.2"; "G.504" -> "G.2"; "G.4" -> "G.3"; "G.504" -> "G.3"; "G.7" -> "G.4"; "G.504" -> "G.4"; "G.7" -> "G.5"; "G.504" -> "G.7"; }

Required for

G.507, G.508, G.509, G.510, G.511, G.512, G.513, G.514, G.515, G.517, G.518, G.522, G.523, G.528, G.530, G.531, G.532, G.541, G.542, G.543, G.544, G.545, G.546, G.547, G.549, G.550, 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.577, G.578, G.592, G.594, G.597, G.600, G.601