Proof
Let line touch the circle at , with centre . Suppose
is not perpendicular to ; drop the perpendicular to
at some point . In right triangle
(right-angled at ), is the hypotenuse, so by I.19, . But lies on the tangent , which has no point inside
the circle (Definition III.2); so radius. The two
inequalities contradict. Hence and .
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Full neighborhood
Depends on (4)
- I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
- I.19Proposition I.19In any triangle the greater angle is subtended by the greater side.
- III.2Definition III.2A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.
- I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…
Required by (dependents) (7)
- III.19Proposition III.19If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the…
- III.32Proposition III.32If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line…
- III.36Proposition III.36If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut…
- III.37Proposition III.37If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut…
- IV.3Proposition IV.3About a given circle to circumscribe a triangle equiangular with a given triangle.
- IV.7Proposition IV.7About a given circle to circumscribe a square.
- IV.12Proposition IV.12About a given circle to circumscribe an equilateral and equiangular pentagon.
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