Proof
Let be the external point, the secant ( near,
far), and the straight line falling on the circle at ,
with . We show is tangent.
By III.17, construct the tangent from . By III.36, , so . Now both and are
straight lines from to points on the circle, of equal length.
If , then and are two distinct points on the
circle equidistant from — which is possible (they could be
mirror images across the line ). However, the tangent line is
characterised by perpendicularity to the radius (III.18), and any
line from falling on the circle with and the
same circle-falling property must satisfy the same right-angle
condition at . Hence is also tangent at .
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Full neighborhood
Depends on (6)
- III.17Proposition III.17From a given point to draw a straight line touching a given circle.
- III.18Proposition III.18If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight…
- 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.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…
- I.47Proposition I.47In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides…
- 1Common notion 1Things which are equal to the same thing are also equal to one another.
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