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.

160 A SEQUEL TO EUCLID.
109. A right line, which bisects the perimeter of the maximum
figure contained by that perimeter, bisects also the area of the
figure. Hence show, that of all figures having the same perimeter
a circle has the greatest area.
110. The polar circle of a triangle, its circumscribed circle,
and nine-points circle, are coaxal.
111. The polar circles of the five triangles external to a pentagon, which are foc'med by producing its sides, have a common
orthogonal circle.
112. The six anharmonic ratios of four collinear points can be
expressed in terms of the trigonometrical functions of an angle,
namely,
- sin2o, - cos2p, tan2p, - cosec2p, -sec2p, cot2p.
Show how to construct p.
113. If the sides of a polygon of an even number of sides be
cut by any transversal, the product of one set of alternate segments is equal to the product of the other set. If the number of
sides of the polygon be odd, the rectangles will be equal, but will
have contrary signs (CARNOT).
114. If from the angular points of a polygon of an odd number
of sides concurrent lines be drawn, dividing the opposite sides
each into two segments, the product of one set of alternate segments is equal to the product of the other set (PONCELET).
115. If the points at infinity on two lines divided homographically be corresponding points, the lines are divided proportionally.
116. To construct a quadrilateral, beizg given thefour sides and
the area.
Analysis. -Let ABCD be C B
the required quadrilateral. The
four sides, AB, BC, CD, DA are
given in magnitude; and the
area is also given. Draw AE
parallel and equal to BD. Join
ED, EC; draw AF, CG per- A
pendicular to BD; produce CG D
to H; bisect BD in 0.
Now we have
BC2 - CD2 = 2BD. OG
and
AD2- AB2 = 2BD, OF;