Proof
Take any other straight line through the foot in the plane;
can be expressed as a sum of perpendicular components on the
two given lines (by I.46-style decomposition), and the perpendicular
to both is perpendicular to the sum by I.4 applied to the right
triangles formed.
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Depends on (3)
- I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight…
- I.8Proposition I.8If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they…
- XI.3Definition XI.3A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and…
Required by (dependents) (6)
- XI.5Proposition XI.5If a straight line be set up at right angles to three straight lines which meet one another, at their common point of…
- XI.6Proposition XI.6If two straight lines be at right angles to the same plane, the straight lines will be parallel.
- XI.8Proposition XI.8If two straight lines be parallel, and one of them be at right angles to any plane, the remaining one will also be at…
- XI.11Proposition XI.11From a given elevated point to draw a straight line perpendicular to a given plane.
- XI.13Proposition XI.13From the same point two straight lines cannot be set up at right angles to the same plane on the same side.
- XI.18Proposition XI.18If a straight line be at right angles to any plane, all the planes through it will also be at right angles to the same…
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