G.119 draw a tangent to a circle from a point — Heath

III.17

From a given point to draw a straight line touching a given circle.

Let A be the given point, and ( B C D ) the given circle; thus it is required to draw from the point A a straight line touching the circle ( B C D ).

For let the centre E of the circle be taken; [G.98] let / A E / be joined, and with centre E and distance / E A / let the circle ( A F G ) be described; from D let / D F / be drawn at right angles to / E A /, and let / E F /, / A B / be joined; I say that / A B / has been drawn from the point A touching the circle ( B C D ).

For, since E is the centre of the circles ( B C D ), ( A F G ),

/ E A / is equal to / E F /, and / E D / to / E B /;

therefore the two sides / A E /, / E B / are equal to the two sides / F E /, / E D /: and they contain a common angle, the angle at E;

therefore the base / D F / is equal to the base / A B /, and the triangle < D E F > is equal to the triangle < B E A >, and the remaining angles to the remaining angles; [G.24] therefore the angle < E D F ) is equal to the angle < E B A ).

But the angle < E D F ) is right;

therefore the angle < E B A ) is also right.

Now / E B / is a radius; and the straight line drawn at right angles to the diameter of a circle, from its extremity, touches the circle; [G.118]

therefore / A B / touches the circle ( B C D ).

Therefore from the given point A the straight line / A B / has been drawn touching the circle ( B C D ).

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