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Proposition 16

Tue, 2010-02-16 08:05 — ashley

The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle.

Book 3 Proposition 16

Let $ ABC $ be a circle about $ D $ as centre and $ AB $ as diameter;

I say that the straight line drawn from $ A $ at right angles to $ AB $ from its extremity will fall outside the circle.

For suppose it does not, but, if possible, let if fall within as $ CA $ and let $ DC $ be joined.

Since $ DA $ is equal to $ DC $,

the angle $ DAC $ is also equal to the angle $ ACD $ [Prop 1.5]

But the angle $ DAC $ is right;

therefore the angle $ ACD $ is also right:

thus, in the triangle $ ACD $, the two angles $ DAC $, $ ACD $ are equal to two right angles: which is impossible. [Prop 1.17]

Therefore the straight line drawn from the point $ A $ at right angles to $ BA $ will not fall within the circle.

Similarly we can prove that neither will it fall on the circumference;

therefore it will fall outside.

Let it fall as $ AE $.

I say next that into the space between the straight line $ AE $ and the circumference $ CHA $ another straight line cannot be interposed.

For, if possible, let another straight line be so interposed, as $ FA $, and let $ DG $ be drawn from the point $ D $ per pendicular to $ FA $.

Then, since the angle $ AGD $ is right,

and the angle $ DAG $ is less than a right angle,

$ AD $ is greater than $ DG $. [Prop 1.19]

But $ DA $ is equal to $ DH $;

therefore $ DH $ is greater than $ DG $, the less to the greater : which is impossible.

Therefore another straight line cannot be interposed into the space between the straight line and the circumference.

I say further that the angle of the semicircle contained by the straight line $ BA $ and the circumference $ CHA $ is greater than any acute rectilineal angle,

and the remaining angle contained by the circumference $ CHA $ and the straight line $ AE $ is less than any acute rectilineal angle.

For, if there is any rectilineal angle greater than the angle contained by the straight line $ BA $ and the circumference $ CHA $, and any rectilineal angle less than the angle contained by the circumference $ CHA $ and the straight line $ AE $, then into the space between the circumference and the straight line $ AE $ a striaght line will be interposed such as will make an angle contained by straight lines which is greater than the angle contained by the straight line $ BA $ and the circumference $ CHA $, and another angle contained by straight lines which is less than the angle contained by the circumference $ CHA $ and the straight line $ AE $.

But such a straight line cannot eb interposed;

therefore there will not be any acute angle contained by straight lines which is greater than the angle contained by the straight line $ BA $ and the circumference $ CHA $, nor yet any acute angle contained by straight lines which is less than the angle contained by the circumference $ CHA $ and the straight line $ AE $. --

$ {\small {\tt PORISM}} $. From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its extremity touches the circle.

Q.E.D.