34 CHAPTER 1 Advanced Euclidean Geometry
- Prove the “Explicit Law of Sines,”
namely that if we are given the tri-
angle 4 ABCwith sidesa, b, andc,
and ifRis the circumradius, then
a
sinA=
b
sinB=
c
sinC= 2R.
Conclude that the perimeter of the
triangle is
a+b+c= 2R(sinA+ sinB+ sinC).- Let a circle be given with centerOand radiusr. LetP be a given
point, and let d be the distance OP. Letl be a line through P
intersecting the circle at the pointsAandA′. Show that
(a) IfP is inside the circle, thenPA×PA′=r^2 −d^2.(b) IfP is outside the circle, thenPA× PA′=d^2 −r^2.Therefore, if we use sensed magnitudes in defining the power ofP
relative to the circle with radiusr, then the power ofP relative to
this circle is alwaysd^2 −r^2.- Given the circle C and a real numberp, describe the locus of all
pointsP having powerprelative toC. - LetP be a point and letCbe a circle. LetAandA′beantipodal
points on the circle (i.e., the line segment [AA′] is a diameter of
C). Show that the power ofP relative toCis given by the vector
dot product
−→
PA·−→
PA′. (Hint: Note that ifO is the center of C,
then−→
PA=−→
PO+−→
OAand−→
PA′=−→
PO−−→
OA. Apply exercise 3.)