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.

ADDITIONAL PROPOSITIONS AND EXERCISES. 161
therefore
BC2 + AD2 - AB - CD2 = 2BD. FG = 2AE. AH.
Hence, since the four lines AB, BC, CD, DA are given in
magnitude, the rectangle AE. AH is given. Now, if we suppose
the line AD to be given in position, since DE is equal to AB.
which is given in magnitude, the locus of the point E is a circle,
and since the rectangle AH. AE is given, the locus of the point H
is a circle, namely, the inverse of the locus of E.
Again, since the lines AE, AC are equal, respectively, to the
diagonals of the quadrilateral, and include an angle equal to that
between the diagonals, the area of the triangle ACE is equal to
the area of the quadrilateral. Hence the area of the triangle
ACE is given. Therefore the rectangle AE. CH is given. And
it has been proved that the rectangle AE. AH is given; therefore
the ratio AH: CH is given. Hence the triangle ACH is given
in species. And since the point A is fixed, and H moves on a
given circle, C moves on a given circle. And since D is fixed,
and DC given in magnitude, the locus of the point C is another
circle. Hence C is a given point.
117. Prove from the foregoing analysis that the area is a maximum when the four points A, B, C, D are concyclic.
118. In the same case prove that the angle between the diagonals
is a maximum when the points are concyclic.
119. The difference of the squares of the two interior diagonals
of a cyclic quadrilateral is to twice their rectangle as the distance
between their middle points is to the third diagonal.
120. Inscribe in a given circle a quadrilateral whose three
diagonals are given. [Make use of Ex. 119.]
121. Given the two diagonals and all the angles of a quadrilateral; construct it.
122. If L be one of the limiting points of two circles, 0, 0', and
LA, LB two radii vectors at right angles to each other, and terminating in these circles, the locus of the intersection of tangents at A
and B is a circle coaxal with 0, 0'.
Dem.-Join AB, intersecting the circles again in G and H,
and let fall the perpendiculars OC, O'D, LE.
X