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.

64

A SEQUEL TO EUCLID.

SECTION II.
EXERCISESo
1. The square of the side of an equilateral A inscribed in a (
equal three times the square of the radius.
2. The square described about a O equal twice the inscribed
square.
3. The inscribed hexagon equal twice the inscribed equilateral triangle.
4. In the construction of IV., x., if F be the second point in
which the 0 ACD intersects the ( BDE, and if we join AF, DF,
the A ADF has each of its base L s double the vertical L. The
same property holds for the As ACF, BCD.
5. The square of the side of a hexagon inscribed in a (, together
with the square of the side of a decagon, is equal to the square of
the side of a pentagon.
6. Any diagonal of a pentagon is divided by a consecutive
diagonal into two parts, such that the rectangle contained by the
whole and one part is equal to the square of the other part.
7. Divide an L of an equilateral A into five equal parts.
8. Inscribe a ( in a given sector of a circle.
9. The locus of the centre of the 0 inscribed in a A, whose
base and vertical L are given, is a circle.
10. If tangents be drawn to a 0 at the angular points of an
inscribed regular polygon of any number of sides, they will form
a circumscribed regular polygon.
11. The line joining the centres of the inscribed and circumscribed Os subtends at any of the angular points of a A an L
equal to half the difference of the remaining angles.
12. Inscribe an equilateral A in a given square.
13. The six lines of connexion of the centres of the inscribed
and escribed Os of a plane A are bisected by the circumference
of the circumscribed circle.
14. Describe a regular octagon in a given square.
15. A regular polygon of any number of sides has one ( inscribed in it, and another circumscribed about it, and the two (s
are concentric.