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.

BOOK I. 17
but EN = AB, and NK = 2AC;
therefore EK2 = AB2 + 4AC2.
In like manner
FD2 = 4AB2 + AC,;
therefore EK2 + FD2 = 5 (AB2 + AC2) = 5BC2.
(4). The intercepts AQ, AR are equal.
(5). The lines CF, BK, AL are concurrent.
SECTION II.
EXERCISES.
1. The line which bisects the vertical L of an isosceles A bisects
the base perpendicularly.
2. The diagonals of a quadrilateral whose four sides are equal
bisect each other perpendicularly.
3. If the line which bisects the vertical L of a A also bisects the
base, the A is isosceles.
4. From a given point in one of the sides of a A draw a right
line bisecting the area of the A.
5. The sum of the J s from any point in the base of an isosceles
A on the equal sides is = to the J from one of the base angles on
the opposite side.
6. If the point be taken in the base produced, prove that the
difference of the Is on the equal sides is = to the 1 from one of
the base angles on the opposite side; and show that, having
regard to the convention respecting the signs plus and minus, this
theorem is a case of the last.
7. If the base BC of a A be produced to D, the L between the
bisectors of the L s ABC, ACD = half L BAC.
8. The bisectors of the three internal angles of a A are concurrent.
9. Any two external bisectors and the third internal bisector of
the angles of a A are concurrent.
10. The quadrilaterals formed either by the four external or the
four internal bisectors of the angles of any quadrilateral have their
opposite L s = two right L s.
C