Circle Theorems

The vocabulary identifies the configuration

A radius joins the centre to a point on the circle. A chord joins two points on the circle. A diameter is a chord through the centre. A secant is a line meeting the circle at two distinct points, while a tangent meets it at one point.

The same infinite line may have an external segment and a segment inside the circle. When a theorem uses a whole secant length, that length begins at the external point and ends at the far circle intersection. It is not just the part inside the circle.

A perpendicular from the centre bisects a chord

Let $AB$ be a non-diameter chord and let the perpendicular from centre $O$ meet it at $M$. The right triangles $OMA$ and $OMB$ have equal hypotenuses $OA=OB$, because both are radii, and the common leg $OM$. RHS congruence gives $AM=MB$.

Conversely, if $M$ is the midpoint of a non-diameter chord, then $OM$ is perpendicular to the chord. For a diameter, the centre is already its midpoint because the two halves are radii. The centre and midpoint then coincide, so they do not determine a unique line.

Finding a chord length

A circle has radius $5\text{ cm}$, and a chord is $3\text{ cm}$ from the centre. Half the chord is a leg of a right triangle:

The common error is to stop at $4\text{ cm}$, which gives only half the required length. The chord lies inside a diameter of length $10\text{ cm}$, so an $8\text{ cm}$ result is plausible.

In the same circle, or in equal-radius circles, equal chords have equal perpendicular distances from the centre, and conversely. A chord closer to the centre is longer. The same-radius condition matters: equal lengths in different-sized circles do not determine equal central distances.

A tangent is perpendicular to its radius

At a point of tangency $T$, the radius $OT$ is perpendicular to the tangent line. Another line through $T$ that is not perpendicular passes through the circle’s interior and meets the circle again. The perpendicular therefore identifies the tangent direction.

If an external point $P$ is $13\text{ cm}$ from the centre of a radius-$5\text{ cm}$ circle, a tangent segment $PT$ has length

The right angle is at $T$, so $OP$ is the hypotenuse. It is not correct to label the tangent segment as the hypotenuse just because it is outside the circle.

Tangents from one external point are equal

Suppose $PA$ and $PB$ are tangents from the same external point $P$ to one circle. The triangles $OAP$ and $OBP$ are right-angled, have common hypotenuse $OP$ and equal radius legs $OA=OB$. They are congruent by RHS, so

If the lengths are $3x-2$ and $x+6$, then $3x-2=x+6$ gives $x=4$. Each tangent segment is $10$ units. Tangents drawn from two different external points do not generally have equal lengths.

Intersecting chords

If chords $AB$ and $CD$ intersect at an interior point $P$, then

To see why, compare $\triangle APC$ and $\triangle DPB$. Their angles at $P$ are vertically opposite. Their other matching angles are inscribed angles subtending the same arcs. The triangles are similar, so $PA/PD=PC/PB$, which gives the product equation.

If $PA=3$, $PB=8$ and $PC=4$, then $3(8)=4(PD)$ and $PD=6$. Each product uses the two pieces of a single chord, not adjacent segments chosen by appearance.

Secants and the tangent-secant theorem

Let $P$ be outside the circle. Suppose one secant reaches the near point $A$ and then the far point $B$, while another reaches $C$ then $D$. The relationship is

The symbols look like the intersecting-chords equation, but the lengths now mean external part times whole secant. If $PA=4$ and $AB=5$, the whole length is $PB=9$. If $PC=3$, then $3(PD)=4(9)=36$, so $PD=12$ and the interior part $CD=9$.

For a tangent $PT$ and a secant through $A,B$ from the same external point,

With external part $4$ and interior part $5$, the tangent length is $\sqrt{4(9)}=6$. This statement concerns segment lengths, not angles. The AMSI circle module develops these relationships through similarity.

Choosing a theorem from the givens

First identify whether the known point lies inside, on or outside the circle. Next label the complete lines and the segment endpoints. For a chord-distance problem, draw the perpendicular radius to expose a right triangle. For two external lines, mark the near and far intersections before forming a product.

Treat the sketch as a record of the hypotheses. An apparently tangent line needs a stated tangency condition or a proof that it is perpendicular to the radius at a boundary point. A diagram touching the curve on screen is not itself evidence.

Related topics

Angles in Circles explains the inscribed-angle facts used in the similarity argument. Chords, Arcs and Sectors separates straight chord length from curved arc length and region area.