Sidebilder
PDF
ePub

F G is not perpendicular to DÉ. And in the same Manner we prove, that no other Right Line but FC is perpendicular to DE. Wherefore F C is perpendicular to D E. Therefore, if any Right Line touches a Circle, and from the Center to the Point of Contact a Right Line be drawn; that Line will be perpendicular to the Tangent; which was to be demonstrated.

PROPOSITION XIX.

THE OR EM.
If any Right Line touches a Circle, and from the

Point of Contakt a Right Line be drawn at
Right Angles to the Tangent, the Center of the

Circle shall be in the said Line.
L

ET any Right Line D E touch the Circle ABC

in C, and let CA be drawn from the Point C at Right Angles to DE. I say, the Circle's Center is in AC.

For if it be not, let F be the Center, if possible, and join CF.

Then because the Right Line DE touches the Circle ABC, and FC is drawn from the Center to

the Point of Contact ; FC will be perpendicular to . 18 of this. DE* And so the Angle FCE is a Right one. From the But ACE is also a Right Anglet: Therefore the

Angle FCE is equal to the Angle ACE, a less to a greater; which is absurd. Therefore F is not the Center of the Circle ABC. After this Manner we prove, that the Center of the Circle can be in no other Line, unless in AC. Wherefore, if any Right Line touches a Circle, and from the Point of Contact a Right Line be drawn at Right Angles to the Tangent, the Center of the Circle Mall be in the said Line ; which was to be demonstrated.

i

[merged small][merged small][ocr errors]
[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

F G is not perpendicular to DÉ. And in the same Manner we prove, that no other Right Line but FC is perpendicular to DE. Wherefore F C is perpendicular to DE. Therefore, if any Right Line touches a Circle, and from the Center to the point of Contatt a Right Line be drawn; that Line will be perpendicular to the Tangent; which was to be demonstrated.

PROPOSITION XIX.

[ocr errors]

THEOREM.
If any Right Line touches a Circle, and from the

Point of Contact a Right Line be drawn at
Right Angles to the Tangent, the Center of the
Circle shall be in the said Line.
ET any Right Line DE touch the Circle ABC

in C, and let CA be drawn from the Point
at Right Angles to DE. I say, the Circle's Center is
in AC.

For if it be not, let F be the Center, if poffible, and join CF.

Then because the Right Line DE touches the Circle ABC, and FC is drawn from the Center to

the Point of Contact ; FC will be perpendicular to * 18 of this. DE*. And so the Angle FCE is a Right one. † From tbe But ACE is also a Right Anglet: Therefore the Нур.

Angle FCE is equal to the Angle A CE, a less to a greater; which is absurd. Therefore F is not the Center of the Circle ABC. After this Manner we prove, that the Center of the Circle can be in no other Line, unless in A C. Wherefore, if any Right Line touches a Circle, and from the Point of Conta&t a Right Line be drawn at Right Angles to the Tangent, the Center of the Circle fall be in the said Line; which was to be demonstrated.

[merged small][ocr errors]
[merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]
[ocr errors][ocr errors]
« ForrigeFortsett »