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.

50

A SEQUEL TO EUCLID.

16. Given the base of a A and the vertical L, what is the
locus-(1) of the intersection of the is; (2) of the bisectors of
the base angles?
17. Of all As inscribed in a given 0, the equilateral A is a
maximum.
18. The square of the third diagonal of a quadrilateral inscribed in a ( is equal to the sum of the squares of tangents to
the ( from its extremities.
19. The 0, whose diameter is the third diagonal of a quadri.
lateral inscribed in another 0, cuts the latter orthogonally.
20. If from any point in the circumference of a ( three lines
be drawn to the angular points of an inscribed equilateral A, one
of these lines is equal to the sum of the other two.
21. If the feet of the I of a A be joined, the A thus formed
will have its angles bisected by the i s of the original triangle.
22. If all the sides of a quadrilateral or polygon, except one, be
given in magnitude and order, the area will be a maximum, when
the remaining side is the diameter of a semicircle passing through
all the vertices.
23. The area will be the same in whatever order the sides are
placed.
24. If two quadrilaterals or polygons have their sides equal,
each to each, and if one be inscribed in a 0, it will be greater
than the other.
25. If from any point P without a 0 a secant be drawn cutting the ( in the points A, B; then if C be the middle point of
the polar of P, the L ACB is bisected by the polar of P.
26. If OPP' be any line cutting a 0, J, in the points PP'; then
if two Os passing through O touch J in the points P, P', respectively, the difference between their diameters is equal to the diameter of J.
27. Given the base, the difference of the base L s, and the sum
or difference of the sides of a A, construct it.
28. Given the base, the vertical L, and the bisector of the
vertical L of a A, construct it.
29. Draw a right line through the point of intersection of two
Os, so that the sum or the difference of the squares of the intereepted segments shall be given.
30. If an arc of a 0 be divided into two equal, and into two
unequal parts, the rectangle contained by the chords of the unequal parts, together with the square of the chord of the arc between the points of section, is equal to the square of the chord of
half the arc.