Proof: By Euclid
(related to Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line)
∎
- For let the two planes $AB$ and $BC$ cut one another, and let their common section be the line $DB$.
- I say that the line $DB$ is straight.
- For, if not, let the straight line $DEB$ have been joined from $D$ to $B$ in the plane $AB$, and the straight line $DFB$ in the plane $BC$.
- So two straight lines, $DEB$ and $DFB$, will have the same ends, and they will clearly enclose an area.
- The very thing (is) absurd.
- Thus, $DEB$ and $DFB$ are not straight lines.
- So, similarly, we can show than no other straight line can be joined from $D$ to $B$ except $DB$, the common section of the planes $AB$ and $BC$.
- Thus, if two planes cut one another then their common section is a straight line.
- (Which is) the very thing it was required to show.
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Footnotes
