A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

34

A SEQUEL TO EUCLID.

the diagonals of an inscribed quadrilateral, the rectangle
contained by the perpendiculars on either pair of opposite
sides is equal to the
rectangle contained by F
the perpendiculars on /
the diagonals./
Dem. —Let OE, OF D
bethe -s ontheoppo-
site sides AB, CD; OG,
0H, the ILs on the
diagonals. Join EG,
FH, OA, OD. Now,B\
as in the last Prop., we
see that the quadrilaterals AEOG, DFOH,
are inscribed in Os.
Hence Z OEG = OAG,
and OHF = ODF. Again, since AODC is a quadrilateral in a 0, the Z OAC + ODC = two right Zs
(xxii.) = ODC + ODF;.-. the Z OAC= ODF. Hence
the Z OEG = OHF. In like manner, the Z OGE
= OFH. Hence the 4s OEG, OHF are equiangular,
and the rectangle OE. OF = the rectangle OG. OH.
Cor. 1.-The rectangle contained by the is on one
pair of opposite sides is equal to the rectangle contained
by the is on the other pair of opposite sides. This may
be proved directly, or it follows at once from the theorem
in the text.
Cor. 2. —If we suppose the points A, B, to become
consecutive, and also the points 0, D, then AB, CD
become tangents; and from the theorem of this Article
we may infer the theorem of Prop. 10.
Prop 12.-The feet D, E, F of the three perpendiculars
let fall on the sides of a triangle ABC, from any point P in
the circumference of the circumscribed circle, are collinear.
Dem.-Join PA, PB, DF, EF. As in the Demonstrations of the two last Propositions, we see that the quadrilaterals PBDF, PFAE are inscribed in Os;.-. the
Z s PBD, PFD are = two right Z s (xxii.), andZ s PBD,