Elements of Plane Geometry
Elements of Plane Geometry
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In our eReader you can find the full English version of the book. Read Elements of Plane Geometry Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book Elements of Plane Geometry
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THEOREM 269. In the same circle, or in equal circles, if two arcs are equal, the chords that subtend them are also equal; and conversely, if two chords are equal, the arcs that are subtended by them are equal.
Let ABC and DEF be two equal arcs in the equal CD whose centers are x and y.
To Prove chord AC = chord DF.
Proof. Draw the radii xA, xC, yD, and yF. Show that Z1=Z2. Prove A AxC and DyF equal.
Whence AC = DF. Q. E. D.
CONVERSELY. Let chord AC = chord DF. To Prove arc ABC = arc DEF. Proof.... Draw the radii xA, xC, yD, and yF. Prove A AxC and DyF equal.
Whence Z1 = Z2.
. -. Arc ABC = arc DEF. (?) Q. E. D.
270. EXERCISE. If the circumference of a circle is divided into four equal parts and their extremities are joined by chords, the resulting quadrilateral is an equilateral parallelogram.
BOOK II 87 PROPOSITION VIII. THEOREM 271. In the same circle, or in equal circles, if two arcs are unequal and each is less than a semi-circumfer- ence, the greater arc is subtended by the greater chord ; and conversely, the greater chord subtends the greater arc.
What to read after Elements of Plane Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Alan Sanders to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
THEOREM 269. In the same circle, or in equal circles, if two arcs are equal, the chords that subtend them are also equal; and conversely, if two chords are equal, the arcs that are subtended by them are equal.
Let ABC and DEF be two equal arcs in the equal CD whose centers are x and y.
To Prove chord AC = chord DF.
Proof. Draw the radii xA, xC, yD, and yF. Show that Z1=Z2. Prove A AxC and DyF equal.
Whence AC = DF. Q. E. D.
CONVERSELY. Let chord AC = chord DF. To Prove arc ABC = arc DEF. Proof.... Draw the radii xA, xC, yD, and yF. Prove A AxC and DyF equal.
Whence Z1 = Z2.
. -. Arc ABC = arc DEF. (?) Q. E. D.
270. EXERCISE. If the circumference of a circle is divided into four equal parts and their extremities are joined by chords, the resulting quadrilateral is an equilateral parallelogram.
BOOK II 87 PROPOSITION VIII. THEOREM 271. In the same circle, or in equal circles, if two arcs are unequal and each is less than a semi-circumfer- ence, the greater arc is subtended by the greater chord ; and conversely, the greater chord subtends the greater arc.
What to read after Elements of Plane Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Alan Sanders to read onlineMoreLess
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