Power Theorem
Given a point P and a circle, pass two lines through P that intersect the circle in
points A and D and, respectively, B and C. Then APDP=BPCP.
P may lie either inside or outside the circle. The line
through A and D (or that through B and C or both) may be tangent to the circle,
in which case A and D coalesce into a single point. In all the cases, the theorem
The point
holds and is known as the Power of a Point Theorem.
When the point
P is inside the circle, the theorem is also known as the Theorem
of Intersecting Chords (or theIntersecting Chords Theorem) and has
a beautiful interpretation . When the point
P is outside the circle, the theorem
becomes the Theorem of Intersecting Secants (or the Intersecting Secants
Theorem.)
The proof is exactly the same in all three cases mentioned above. Since
triangles ABP and CDP are similar, the following equality holds:
APCP=BPDP,
APDP=BPCP.
The common value of the products then depends only on P and the circle and is
known as the Power of Point P with respect to the (given) circle. Note that,
when P lies outside the circle, its power equals the length of the square of the
tangent from P to the circle. For example, if B=C so that BP is tangent to the
circle APDP=BP2.
which is equivalent to the statement of the theorem:
Sometimes it is useful to employ signed segments . The convenience is that it is
possible to tell points inside the circle from the points outside the circle. The
power of a point inside the circle is negative, whereas that of a point outside the
circle is positive. This is exactly what one obtains from the algebraic
definition of the power of a point.
A, B, C, and D are not collinear.
Let P be the intersection of AD and BC such that APDP=BPCP. Then the four
points A, B, C, and D are concyclic. To see that draw a circle through,
say, A, B, and C. Assume it intersects AP at D. Then, as was shown
above, APDP=BPCP,from which D=D. (If, say, B and C coincide, draw the
circle through A tangent to PB at B.)
The theorem is reversible: Assume points
Power of a Point Theorem:
Given circle O, point P not on the circle, and a line through P intersecting the circle
in two points. The product of the length from P to the first point of intersection and
the length from Pto the second point of intersection is constant for any choice of a
line through P that intersects the circle. This constant is called the "power of
point P".
If P is outside the circle ....
If P is inside the circle ....
This becomes the theorem we know as the
theorem of intersecting secants.
Special Cases:
This becomes the theorem we know
as the theorem of intersecting chords.
Should both of the lines be tangents
Should one of the lines be tangent to the
to the circle, point A coincides with
circle, point A will coincide with point D, and point D, point C coincides with point
the theorem still applies.
B, and the theorem still applies.
This becomes the theorem we know as the
theorem of secant-tangent theorem.
This becomes the theorem we know
as the theorem of two tangents.
1.
1. Given
2.
2. Two points determine
exactly one line.
3.
3. Reflexive Property
(Identity)
4. In a circle, the measure
of an inscribed angle is
one-half the measure of
its intercepted arc.
4.
5.
5. Substitution (or
Transitive)
6.
6. Congruent angles are
angles of equal measure.
7.
8.
9.
7. AA (If two angles of
one triangle are congruent
to the corresponding
angles of another triangle,
the triangles are similar.
8. Corresponding sides of
similar triangles are in
proportion.
9. In a proportion, the
product of the means
equals the product of the
extremes.
Theorem: For all
Proof:
You can use the Secant-Secant Power Theorem to solve some circle problems. This
theorem involves are you sitting down two secants! (If youre trying to come up
with a creative name for your child like Dweezil or Moon Unit, talk to Frank Zappa, not
the guy who named the power theorems.)
Secant-Secant Power Theorem: If two secants are drawn from an external point to a
circle, then the product of the measures of one secants external part and that entire
secant is equal to the product of the measures of the other secants external part and
that entire secant. (Whew!)
For instance, in the above figure,
4(4 + 2) = 3(3 + 5)
The following problem uses two power theorems:
Given: Diagram as shown
Segment BA is tangent to circle H at A
Find: x and y
The figure includes a tangent and some secants, so look to your Tangent-Secant and
Secant-Secant Power Theorems.
Now use the Secant-Secant Power Theorem with secants segment EC and
segment EG to solve for y:
A segment cant have a negative length, so y = 3. That does it.
POWER THEOREM
Theorem: For all ,
Proof:
As mentioned in 5.8, physical power is energy per unit time.7.19 For example,
when a forceproduces a motion, the power delivered is given by the force times
the velocity of the motion. Therefore, if and
are in physical units of force and
velocity (or any analogous quantities such as voltage and current, etc.), then their
product
and
is proportional to the power per sample at time ,
becomes proportional to the totalenergy supplied (or absorbed) by the
driving force. By the power theorem,
can be interpreted as
the energy per bin in the DFT, or spectral power, i.e., the energy associated with
a spectral band of width
\Circle
.7.20
Power
The power of a fixed point
with respect to a circle of radius and center
is defined by the product
(1)
where
and
are the intersections of a line through
with the circle. The term "power" was first used in this way by
Jacob Steiner (Steiner 1826; Coxeter and Greitzer 1967, p. 30). Amazingly,
is independent of the choice of the line
(Coxeter 1969, p. 81).
(sometimes written
Now consider a point not necessarily on the circumference of the circle. If
the circle's center , then the power of the point relative to the circle is
is the distance between
and
(2)
If is outside the circle, its power is positive and equal to the square of the length of the segment
tangent to the circle through ,
from
to the
(3)
If
lies along the x-axis, then the angle
around the circle at which
lies is given by solving
(4)
for , giving
(5)
for coordinates
(6)
The points
and
are inverse points, also called polar reciprocals, with respect to the inversion circle if
(7)
(Wenninger 1983, p. 2).
If
is inside the circle, then the power is negative and equal to the product of the diameters through
The powers of circle of radius
a reference triangle are
with center having trilinear coordinates
with respect to the vertices of
(8)
(9)
(10)
(P. Moses, pers. comm., Jan. 26, 2005). The circle function of such a circle is then given by
(11)
The locus of points having power with regard to a fixed circle of radius is a concentric circle of radius
The chordal theorem states that the locus of points having equal power with respect to two given
nonconcentric circles is a line called theradical line
We already know that in a circle the measure of a central angle is equal to the
measure of the arc it intercepts. But what if the central angle had its vertex
elsewhere?
An angle whose vertex lies on a circle and whose sides intercept the circle (the
sides contain chords of the circle) is called an inscribed angle. The measure of
an inscribed angle is half the measure of the arc it intercepts.
Figure %: The inscribed angle measures half of the arc it intercepts
If the vertex of an angle is on a circle, but one of the sides of the angle is
contained in a line tangent to the circle, the angle is no longer an inscribed angle.
The measure of such an angle, however, is equal to the measure of an inscribed
angle. It is equal to one-half the measure of the arc it intercepts.
Figure %: An angle whose sides are a chord and a tangent segment
The angle ABC is equal to half the measure of arc AB (the minor arc defined
by points A and B, of course).
An angle whose vertex lies in the interior of a circle, but not at its center, has
rays, or sides, that can be extended to form two secant lines. These secant lines
intersect each other at the vertex of the angle. The measure of such an angle is
half the sum of the measures of the arcs it intercepts.
Figure %: An angle whose vertex is in the interior of a circle
The measure of angle 1 is equal to half the sum of the measures of arcs AB
and DE.
When an angle's vertex lies outside of a circle, and its sides don't intersect with
the circle, we don't necessarily know anything about the angle. The angle's sides,
however, can intersect with the circle in three different ways. Its sides can be
contained in two secant lines, one secant line and one tangent line, or two
tangent lines. In any case, the measure of the angle is one-half the difference
between the measures of the arcs it intercepts. Each case is pictured below.
Figure %: An angle whose vertex lies outside of a circle
In part (A) of the figure above, the measure of angle 1 is equal to one-half the
difference between the measures of arcs JK and LM. In part (B), the measure of
angle 2 is equal to one-half the difference between the measures of arcs QR and
SR. In part (C), the measure of angle 3 is equal to one-half the difference
between the measures of arcs BH and BJH. In this case, J is a point labeled just
to make it easier to understand that when an angle's sides are parts of lines
tangent to a circle, the arcs they intercept are the major and minor arc defined by
the points of tangency. Here, arc BJH is the major arc.
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Maria Alexandria Etong Orquillas
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Mike Jason Ontua
Kim Sevillano
Maria Alexandria Etong Orquillas
G10-Diamond
