This is illustrated in the diagram above, left.
The above diagram to the right shows that two
tangents through a common point P outside the circle
produce line segments PA and PB of equal length.
This follows from the fact that the two triangles produced are both right triangles of the same height with
a shared hypotenuse, and are hence congruent. Thus:
If PA and PB are tangents to a circle at
points A and B, respectively, then PA and
PB have the same length.
2. Inscribed-Angle Theorems
In the diagram below, left, angles α (the peripheral
angle) and β (the central angle) are subtended by the
same ARC. Thus we have:
For angles subtended by the same arc, the
central angle is always twice that of the
peripheral angle.
This is proved by drawing a radius from the center O to the point at which angle α lies to create two
isosceles triangles. Following the left-hand side of the
next diagram, and noting that the interior angles of a
triangle sum to 180°, we thus have x + y = α and
(180 – 2x) + (180 – 2y) + β = 360, from which it follows that β = 2α. A modification of this argument
shows that the result is still true even if the peripheral
angle is located as shown in the right-hand side of
the diagram, or if the arc under consideration is
more than half the PERIMETER of the circle.
The next three results follow (see diagram
below, right):
i. All angles inscribed in a circle subtended by the
same arc are equal,
ii. All angles inscribed by a diameter are right
angles. (This is known as the theorem of Thales.)
circle theorems 77
Tangent theorems
Central and peripheral angles
Proving the central-angle/peripheral-angle theorem
Consequences of the central-angle/peripheral-angle theorem
The above diagram to the right shows that two
tangents through a common point P outside the circle
produce line segments PA and PB of equal length.
This follows from the fact that the two triangles produced are both right triangles of the same height with
a shared hypotenuse, and are hence congruent. Thus:
If PA and PB are tangents to a circle at
points A and B, respectively, then PA and
PB have the same length.
2. Inscribed-Angle Theorems
In the diagram below, left, angles α (the peripheral
angle) and β (the central angle) are subtended by the
same ARC. Thus we have:
For angles subtended by the same arc, the
central angle is always twice that of the
peripheral angle.
This is proved by drawing a radius from the center O to the point at which angle α lies to create two
isosceles triangles. Following the left-hand side of the
next diagram, and noting that the interior angles of a
triangle sum to 180°, we thus have x + y = α and
(180 – 2x) + (180 – 2y) + β = 360, from which it follows that β = 2α. A modification of this argument
shows that the result is still true even if the peripheral
angle is located as shown in the right-hand side of
the diagram, or if the arc under consideration is
more than half the PERIMETER of the circle.
The next three results follow (see diagram
below, right):
i. All angles inscribed in a circle subtended by the
same arc are equal,
ii. All angles inscribed by a diameter are right
angles. (This is known as the theorem of Thales.)
circle theorems 77
Tangent theorems
Central and peripheral angles
Proving the central-angle/peripheral-angle theorem
Consequences of the central-angle/peripheral-angle theorem