圓心至弦的垂線(Perpendiculars to Chords)
圓心至弦的垂線平分弦(⊥from centre to chord bisects chord）
$$\begin{array}{l}{\text{圓心至弦的垂線平分弦}}\\{\text{若 ON}\perp\mathrm{AB},\text{則 AN}=\mathrm{NB}}\\\end{array}$$
圓心至弦中點的連線垂直弦(line joining centre to mid-pt. of chord ⊥ chord)
$$\begin{array}{l}{\text{圓心至弦中點的連線垂直弦}}\\{\text{若 AN=NB,則 ON AB}}\\\end{array}$$
弦的垂線平分線穿過圓心( ⊥ bisector of chord passes through centre)
$$\begin{aligned}
&\text{弦的垂直平分線穿過圓心} \\
&若CN\bot AB和AN=NB,則 \\
&CN穿過圓心O
\end{aligned}$$
圓形等弦(equal chords of a circle)
等弦心距對等弦(chords euidistant from centre are equal)
$$\begin{array}{l}{\text{與圓心等距的弦等長}}\\{\text{若 OM=ON,則 AB=CD}}\\\end{array}$$
等弦對等弦心距(equal chords, equidistant from centre)
$$\begin{aligned}&\text{等弦與圓心等距}\\&\text{若 AB=CD, 則 OM=ON}\end{aligned}$$
圓的角(Relationship between Angle in Circle)
圓心角兩倍於圓周角 (∠ at centre twice ∠ at circumference)
$$\begin{array}{l}{\text{圓心角兩倍於圓周角}}\\{\mathrm{x=2y}}\\\end{array}$$
半圓上的圓周角(∠ in semi-circle)
$$\begin{aligned}
&\text{半圓上的圓周角} \\
&若AB為直徑,則\angle APB= \\
&\text{90°(相反亦然)}
\end{aligned}$$
同弓形內的圓周角(∠s in the same segment)
$$\begin{array}{l}{\text{同弓形内的圓周角}}\\{\mathrm{x=y~;~a=b}}\\\end{array}$$
等角等弧等弦(Equal Relations among Angles, Arcs and Chords)
等角等弧（Equal angles, equal arcs）
$$\begin{aligned}
&等角對等弧/等弧對等角 \\
&\text{若 x=y,則}\widehat{AB}=\widehat{CD} \\
&\text{(相反亦然)}
\end{aligned}$$
等角等弦（Equal angles, equal chords）
$$\begin{aligned}
&\text{等角對等弦/等弦對等角} \\
&\text{若 x=y, 則AB=CD} \\
&\text{(相反亦然)}
\end{aligned}$$
等弧等弦（Equal arcs, equal chords）：
$$\begin{aligned}
&\text{等弦對等弧/等弧對等弦} \\
&\text{若AB}=\mathrm{CD}, \\
&\text{則}\widehat{AB}=\widehat{CD}(\text{相反亦然})
\end{aligned}$$
弧長與圓心角成比例(arcs prop. to ∠s at centre)
$$\begin{array}{l}{\text{弧與圓心角成比例}}\\{\widehat{AB}:\widehat{CD}=\mathrm{m}:\mathrm{n}}\\\end{array}$$
弧長與圓周角成比例(arcs prop. to ∠s at circumference)
$$\begin{array}{l}{\text{弧與圓周角成比例}}\\{\widehat{AB}:\widehat{CD}=\mathrm{x}:\mathrm{y}}\\\end{array}$$
圓內接四邊形相關的特性(Properties related to Cyclic quadrilateral)
圓內接四邊形對角 (opp.∠s, cyclic quad.)
$$\begin{aligned}&\text{圆内接四邊形對角}\\&\angle A+\angle C=180^{\circ}\text{和}\\&\angle B+\angle D=180^{\circ}\end{aligned}$$
對角互補 (opp ∠s supp.)
$$\begin{aligned}
&\text{對角互補} \\
&\angle A+\angle C=180^{\circ}或 \\
&\angle B+\angle D=180^{\circ},\text{則 A,B,C} \\
&\text{和D共圆。}
\end{aligned}$$
圓內接四邊形外角 (ext.∠s, cyclic quad.)
$$\begin{array}{l}{\text{圓内接四邊形外角}}\\{\mathrm{x=y}}\\\end{array}$$
外角=內對角 (ext. ∠ = int. opp. ∠)
$$\begin{aligned}&\text{外角=内對角}\\&\text{若 y=x,}\\&\text{則 A,B,C 和D 共圓}\end{aligned}$$
同弓形內的圓周角的逆定理(Converse of ∠s in the same segment)
$$\begin{array}{l}{\text{同弓形内的圓周角的逆定理}}\\{\text{若 x=y, 則 A,B,C 和 D 共圓}}\\\end{array}$$
圓切線特性 Tangent properties of circle
切線⊥半徑 / 切線⊥半徑的逆定理 (tangent ⊥ radius)
$$\begin{aligned}
&切線⊥半徑/切線⊥半徑的 \\
&\text{逆定理} \\
&\text{若 PQ 切圓於 T,} \\
&\text{則 PQ}\perp\mathrm{OT}_{\circ}(\text{相反亦然})
\end{aligned}$$
切線性質 Tangent Properties
$$\begin{aligned}
&切線性質 \\
&\text{若PA切圓於A,和PB切圓} \\
&\text{於B,則} \\
&\mathrm{(i)~PA=PB} \\
&(\mathrm{ii})~x=y \\
&\mathrm{(iii)~m=n}
\end{aligned}$$
交錯弓形上的圓周角 / 交錯弓形上的圓周角的逆定理 (∠ in alt. segment)
$$\begin{aligned}&\text{若PQ切圓於T,那麼:}\\\\&\angle\mathrm{QTA}=\angle\mathrm{TBA}\\\\&\angle\mathrm{PTB}=\angle\mathrm{TAB}\end{aligned}$$