SECTION 1.5 Nine-Point Circle 43
ratio. Show thatAB is the harmonic mean ofAX andAY.^9- The figure to the right depicts two
circles having an orthogonal in-
tersection. (What should this
mean?) Relative to the diagram to
the right (OandO′are the centers),
show thatA, C, B, andDare in a
harmonic ratio. - The figure to the right shows a
semicircle with center O and di-
ameter XZ. The segment [PY]
is perpendicular to [XZ] and the
segment [QY] is perpendicular to
[OP]. Show that PQ is the har-
monic mean ofXY andY Z.
O
QPX Y Z1.5 The Nine-Point Circle
One of the most subtle mysteries of Euclidean geometry is the existence
of the so-called “nine-point circle,” that is a circle which passes through
nine very naturally pre-prescribed points.
To appreciate the miracle which this presents, consider first that
arranging for a circle to pass through three noncollinear points is, of
course easy: this is the circumscribed circle of the triangle defined by
these points (and having center at the circumcenter). That a circle will
not, in general pass through four points (even if no three are colinear)
(^9) The harmonic mean occurs in elementary algebra and is how one computes the average rate at
which a given task is accomplished. For example, if I walk to the store at 5 km/hr and walk home
at a faster rate of 10 km/hr, then the average rate of speed which I walk is given by
2 × 5 × 10
5 + 10 =
20
3 km/hr.