Circle Geometry: Chapter Review

Vocabulary and notation

Circle, radius, chord, and diameter

A circle is the boundary of points at positive distance $r$ from center $O$. A chord joins two distinct boundary points; a diameter is a chord through $O$.

Concentric, congruent, and coincident circles

Concentric circles share a center. Congruent circles have equal radii. Coincident circles have both the same center and the same radius, so they are one point set.

Central angle and arc notation

A central angle has vertex $O$. Arc measure records degrees, not length. Two letters select an unambiguous minor arc; a third point, as in $\widehat{ACB}$, selects a route.

Minor arc, major arc, and semicircle

A minor arc measures between $0^\circ$ and $180^\circ$; a major arc between $180^\circ$ and $360^\circ$. Diameter endpoints bound two semicircles of $180^\circ$.

Secant and tangent

A secant line meets a circle twice; a tangent line meets it exactly once at its contact point. A tangent segment is finite, from a strict external point to contact.

Common external or internal tangent

A line tangent to two circles is external when their centers lie on the same side of it, internal when they lie on opposite sides.

Inscribed angle and intercepted arc

An inscribed angle has three distinct circle points: its vertex and two chord endpoints. Its intercepted arc joins the endpoints and excludes the vertex.

Cyclic quadrilateral

A convex quadrilateral with four distinct vertices on one circle, named in boundary order. Opposite pairs in $ABCD$ are $A,C$ and $B,D$.

Near and far endpoints

Trace a secant ray from its external vertex: its first intersection is near, its second far. The selected near and far arcs exclude the other pair of endpoints.

External, inside, and whole secant segments

In order $P,A,B$, $PA$ is external, $AB$ is the inside chord, and $PB$ is whole. Unqualified segment symbols denote ordinary lengths, positive for distinct endpoints.

Formulas and theorems

• Circle congruence and similarity

  All positive-radius circles are similar. Equal radii characterize congruence. After translating the original center to the target center, dilate by $k=s/r$.

  Use when Original radius $r>0$ and target radius $s>0$; the dilation is about the target center and maps the whole circle onto the target.

• Central angles and arc addition

  A minor arc equals its smaller central angle in degree measure. The two routes between distinct endpoints total $360^\circ$; a semicircle measures $180^\circ$. Adjacent selected arcs add.

  Use when Add routes with disjoint interiors that join to form the desired arc. Equal degree measure alone does not make arcs on unequal-radius circles congruent.

• Equal chords, arcs, and center distances

  Equal chord lengths, equal minor arc measures, and equal perpendicular center distances are equivalent. Equal minor arcs also give equal major arcs. The nearer chord is longer.

  Use when Same circle or congruent circles. The minor/major wording requires non-diameter chords; equal diameters instead bound equal semicircles and have distance zero. Compare distances only at a fixed radius.

• Perpendicular chord bisection

  A line through $O$ that is perpendicular to $AB$ bisects chord $AB$ and both arcs between its endpoints. Conversely, for a non-diameter chord with midpoint $M$, $OM\perp AB$.

  Use when Positive-radius circle. For a diameter, its midpoint is $O$; the zero segment $OM$ determines no line. A perpendicular diameter still bisects both semicircles.

• Chord length and perpendicular distance

  $$\begin{gathered} h^2+d^2=r^2\\ AB=2h\\ AB=2\sqrt{r^2-d^2} \end{gathered}$$

  Text description

  A circle has center O and chord AB. The perpendicular from O meets AB at midpoint M. A square at M marks the right angle and matching single ticks mark the two equal chord halves. OA is the radius r, OM is the perpendicular distance d, and MB is the half chord h. The chord shown is not a diameter.

  Use when Radius $r>0$, perpendicular center distance $0\le d<r$, and half chord $h$. At $d=0$, use the diameter length $2r$ directly; $d=r$ gives no genuine chord.

• Line-circle incidence and tangency

  A line has zero, one, or two intersections when its perpendicular center distance is greater than, equal to, or less than $r$. At a circle point $T$, a line is tangent exactly when perpendicular to $OT$.

  Use when Positive radius; use the entire line and the radius to the actual contact point. Through a strict external point there are two tangents; through a circle point one; through an interior point none.

• Equal tangent segments

  $$\begin{gathered} PT=PU\\ PT^2=OP^2-r^2 \end{gathered}$$

  Text description

  Point P is strictly outside a circle centered at O. Its tangent segments touch at T above and U below. Radii OT and OU meet those segments at square right-angle marks. A matching single tick on each tangent segment marks equal lengths. OP joins the external point to the center and is the hypotenuse of both right triangles. The lower tangent is dashed as well as a different color.

  Use when Two distinct contacts $T,U$ on the same circle from the same strict external $P$, so $OP>r>0$. The right angles are at contact; $OP$ is the common hypotenuse.

• Two-circle positions and common-tangent counts

  $$\begin{gathered} d>R+r:\ (2,2)\\ d=R+r:\ (2,1)\\ R-r<d<R+r:\ (2,0)\\ d=R-r>0:\ (1,0)\\ 0\le d<R-r:\ (0,0) \end{gathered}$$

  Coincident circles have infinitely many common external tangents and no common internal tangents.

  Use when Radii $R\ge r>0$; $d\ge0$ is center distance. Each pair counts distinct (external, internal) lines. Rows describe separate, external contact, intersecting, internal contact, and strict containment. Containment includes unequal concentric circles; equal centers and equal radii give coincidence.

• Inscribed angles and arc comparisons

  $$\begin{gathered} m\angle AVB=\frac a2 \end{gathered}$$

  Text description

  Distinct points A, V, and B lie on one circle. Chords VA and VB make an angle at V. The heavy arc from A to B runs along the lower part of the circle and excludes V. Its label a names the selected arc measure. The rest of the circle contains V and is not the arc used for this angle.

  Use when Three distinct circle points; $a$ measures arc $AB$ excluding $V$, possibly major. Angles intercepting the same arc are equal; vertices on opposite open arcs between the same endpoints give supplementary angles.

• Diameter and right-angle equivalence

  $AB$ is a diameter if and only if $m\angle ACB=90^\circ$.

  Use when All three distinct points $A,B,C$ lie on the same positive-radius circle. The diameter is opposite the right angle.

• Tangent-chord ray choices

  $$\begin{gathered} m\angle RTA=\frac\theta2\\ m\angle STA=\frac{360^\circ-\theta}{2} \end{gathered}$$

  Text description

  T and A lie on a circle centered at O. The vertical tangent through T has ray TR upward and opposite ray TS downward. A and R are on the same side of the horizontal radius line OT. Chord TA is drawn. The solid minor arc TA lies above, and the dashed major arc returns through the lower part of the circle. A right-angle square marks the tangent and contact radius. The angle using TR is half the minor arc; the angle using TS is half the major arc.

  Use when Tangent rays $TR,TS$ are opposite at $T$. For a non-diameter chord $TA$, $R$ and $A$ lie on the same side of line $OT$; $\theta$ is the minor arc measure. A diameter gives $90^\circ$ for either ray.

• Cyclic opposite and exterior angles

  $$\begin{gathered} m\angle A+m\angle C=180^\circ\\ m\angle B+m\angle D=180^\circ\\ m\angle DCE=m\angle BAD \end{gathered}$$

  Text description

  Four distinct circle points follow the convex boundary order A, B, C, D. The polygon sides join consecutive points. Side BC is extended through C to E, so B, C, E are in that order. A is opposite C and B is opposite D. The exterior angle at C is formed by CD and CE, not by an unspecified extension.

  Use when Convex $ABCD$ in boundary order, four distinct vertices, no three collinear. Either supplementary opposite pair proves cyclicity; so does the exterior equality when $B,C,E$ are collinear in that order. Cyclicity gives all three relations.

• Inside angle: half-sum

  $$\begin{gathered} m\angle APB=\frac{a+c}{2} \end{gathered}$$

  Text description

  Distinct points follow circle order A, B, C, D. Chords AC and BD cross at the strictly interior point P. The solid arc AB faces angle APB. The dashed arc CD faces its vertical angle CPD. These two arcs are the selected pair for the half-sum; the adjacent angle uses the other two arcs.

  Use when Distinct circle points in order $A,B,C,D$; chords $AC,BD$ cross strictly inside at $P$. Measures $a,c$ belong to arcs $AB,CD$ facing this vertical pair, each excluding the other endpoints.

• Two-secant angle: half-difference

  $$\begin{gathered} m\angle APD=\frac{f-n}{2} \end{gathered}$$

  Text description

  P is strictly outside a circle. One ray reaches A first and B second. The other reaches D first and C second. The solid far arc runs from B to C around the right side without passing through A or D. The dashed near arc runs from D to A around the left side without passing through B or C. The angle is between rays PA and PD, both pointing toward the circle.

  Use when Two distinct secant rays from strict external $P$ have orders $P,A,B$ and $P,D,C$. Far arc $BC$ excludes $A,D$ and has measure $f$; near arc $DA$ excludes $B,C$ and has measure $n$. Use rays toward the circle; $f>n$.

• Tangent-secant angle: half-difference

  $$\begin{gathered} m\angle TPA=\frac{f-n}{2} \end{gathered}$$

  Text description

  PT is tangent to the circle at T above the horizontal secant. The secant ray reaches A first and B second from P. The solid near arc TA runs around the upper left without B. The dashed far arc BT runs across the upper right without A. The unhighlighted third arc is AB below the secant and excludes T. The square at T marks the tangent perpendicular to contact radius OT.

  Use when Strict external $P$, tangent contact $T$, and distinct secant intersections in order $P,A,B$. Near arc $TA$ excludes $B$; far arc $BT$ excludes $A$. Their measures are $n,f$. The far route may be major.

• Two-tangent angle

  $$\begin{gathered} m\angle TPU=180^\circ-n \end{gathered}$$

  Use when Strict external $P$; distinct tangent contacts $T,U$. Minor contact arc measure $0^\circ<n<180^\circ$. The major arc is $360^\circ-n$; reversing just one tangent ray gives the supplement. The original angle may be obtuse.

• Inside chord products and correspondence

  $$\begin{gathered} PA\cdot PB=PC\cdot PD\\ \triangle APC\sim\triangle DPB \end{gathered}$$

  Text description

  Chords AB and CD cross strictly inside one circle at P. Their orders are A, P, B and C, P, D. Joining A to C and D to B gives the solid triangle APC and the dashed triangle DPB. The triangle names match A with D, P with P, and C with B. Each product uses the two opposite parts on one straight chord, rather than adjacent parts from different chords.

  Use when Distinct chords in one circle with orders $A,P,B$ and $C,P,D$, $P$ strictly inside. Join $AC,DB$ for the stated AA correspondence. Pair the two positive parts on each straight chord.

• Two-secant products and correspondence

  $$\begin{gathered} PA\cdot PB=PC\cdot PD\\ PB=PA+AB\\ \triangle PAD\sim\triangle PCB \end{gathered}$$

  Text description

  P is strictly outside one circle. The upper secant has order P, A, B and the lower has order P, C, D. A and C are near endpoints; B and D are far. The solid auxiliary chord AD and dashed auxiliary chord CB identify triangles PAD and PCB. Their ordered correspondence matches P with P, A with C, and D with B. PA is external and PB is whole; PC is external and PD is whole.

  Use when Two distinct secant rays to one circle from strict external $P$, with orders $P,A,B$ and $P,C,D$. Join $AD,CB$ for the correspondence. Every product is external times whole, with whole greater than external.

• Tangent-secant product and correspondence

  $$\begin{gathered} PT^2=PA\cdot PB\\ \triangle PTA\sim\triangle PBT \end{gathered}$$

  Use when Tangent contact $T$ and genuine secant order $P,A,B$, from the same strict external point to the same circle. Join $TA,TB$ for the correspondence; all factors are positive.

• Signed power and ordinary products

  $$\begin{gathered} \operatorname{Pow}(P)=OP^2-r^2\\ \text{inside: }PA\cdot PB=-\operatorname{Pow}(P)\\ \text{outside: }PA\cdot PB=\operatorname{Pow}(P) \end{gathered}$$

  Use when Positive radius. Power is negative inside, zero on, positive outside. Inside use opposite chord parts; outside use external and whole secant lengths. Outside it also equals the tangent length squared. On-circle secant products have a zero factor.

  e.g. For radius 14 units and center distance 9 units, find signed power: $-115$ square units.

Problem types, step by step

Exam traps

• Trap Using a half chord as the full answer

  Fix The right triangle contains half the chord. Use the perpendicular center distance, double the half length, and retain the diameter boundary.

• Trap Treating a circle point as a strict external point

  Fix An on-circle point has zero power and one tangent line, with no positive segment to a different tangent contact. Inside points have no tangent through them.

• Trap Choosing the arc containing an inscribed vertex

  Fix Use the other route between the chord endpoints. It can be major; do not automatically halve the minor arc.

• Trap Naming the tangent line but ignoring its ray

  Fix Opposite tangent rays generally give supplementary angles. Reversing one external-angle ray also changes the angle to its supplement.

• Trap Using consecutive angles or diagonal parts in a cyclicity test

  Fix Use whole opposite interior angles with distinct convex boundary-order vertices. Establish the circle before same-arc transfers.

• Trap Calling every exterior angle acute

  Fix Far arcs can be major. Trace near and far endpoints from the vertex, preserve exclusions, and verify positive arcs completing a full circle.

• Trap Multiplying external by inside

  Fix Outside, use external times whole. Inside, use the two opposite parts of one chord. Ordinary inside products are positive, although signed power is negative.

• Trap Rejecting every negative parameter or keeping every root

  Fix Check the expressions that actually represent lengths and their point order. A parameter may be negative; an ordinary positive segment length may not.