Advanced High-School Mathematics

40 CHAPTER 1 Advanced Euclidean Geometry

Exercises

1. [AB] and [AC] are chords of a circle with centerO. X andY are
   the midpoints of [AB] and [AC], respectively. Prove thatO, X, A,
   andY are concyclic points.


2. Derive the Pythagorean Theorem from Ptolemy’s theorem. (This
   is very easy!)


3. Derive Van Schooten’s theorem (see page 35) as a consequence of
   Ptolemy’s theorem. (Also very easy!)


4. Use the addition formula for the sine to prove that ifABCDis a
   cyclic quadrilateral, thenAC·BD=AB·DC+AD·BC.


5. Show that if ABCD is a cyclic quadrilateral with side length
   a, b, c,andd, then the areaK is given by

K =

√

(s−a)(s−b)(s−c)(s−d),

wheres= (a+b+c+d)/2 is the semiperimeter.^8

1.4 Internal and External Divisions; the Harmonic Ratio

The notion ofinternalandexter-
naldivision of a line segment [AB]
is perhaps best motivated by the
familiar picture involving internal
and external bisection of a trian-
gle’s angle (see the figure to the

right). Referring to this figure, we say that the point X divides the
segment [AB] internallyand that the point Y divides the segment
[AB]externally. In general, if A, B, andX are colinear points, we

(^8) This result is due to the ancient Indian mathematician Brahmagupta (598–668).