【Deductive Geometry 演繹幾何 Reasoning】直線│平行線│三角形│四邊形│多邊形│圓形【完整列表】

Deductive Geometry（演繹幾何）是中學數學中最麻煩的課題之一，

涵蓋角度、平行線、三角形、四邊形、多邊形及圓形的基本性質與證明技巧。

無論是 HKDSE，還是一般校內測驗，幾何證明題都經常出現，

而掌握核心定理與 abbreviations（簡寫）更是不掉分的關鍵。

Deductive Geometry 演繹幾何概覽

Angles related to lines & points 直線&點與角
Parallel Lines 平行線
Triangle 三角形
Congruent Triangle 全等三角形
Similar Triangle 相似三角形
Four Centre of Triangle 三角形四心
Parallelogram 平行四邊形
Other Quadrilaterals 其他四邊形
Convex Polygon 凸多邊形
Basic Properties of Circles 圓的基本性質

Angles related to lines & points 直線&點與角

同頂角 (Angles at a point)

Given	a, b and c are angles at a point
Result	a + b + c = 360°
Abbr.	∠s at a pt.

直線上的鄰角 (Adjacent angles on a straight line)

Given	a and b are adjacent angles on a straight line
Result	a + b = 180°
Abbr.	adj. ∠s on st. line

對頂角 (Vertical opposite angles)

Given	Two straight lines AB and CD intersect at point O
Result	a = b and c = d
Abbr.	vert. opp. ∠s

Parallel Lines 平行線

同位角 (Corresponding angles)

Given	AB // CD
Result	a = b
Abbr.	corr. ∠s

Given	a = b
Result	AB // CD
Abbr.	corr. ∠s equal

內錯角 (Alternate angles)

Given	AB // CD
Result	c = d
Abbr.	alt. ∠s

Given	c = d
Result	AB // CD
Abbr.	alt. ∠s equal

同旁內角 (Interior angles)

Given	AB // CD
Result	e + f = 180°
Abbr.	int. ∠s

Given	e + f = 180°
Result	AB // CD
Abbr.	int. ∠s supp.

中點定理 (Mid-point Theorem)

Given	AM = MB and AN = NC
Result	MN = 1/2 BC and MN ∥ BC
Abbr.	mid-pt. thm.

截線定理 (Intercept Theorem)

Given	L₁ ∥ L₂ ∥ L₃ and AB = BC
Result	DE = EF
Abbr.	intercept thm.

Triangle 三角形

三角形內角和 (Angle sum of triangle)

Given	ABC is a triangle
Result	a + b + c = 180°
Abbr.	∠ sum of △

三角形外角 (Exterior angle of triangle)

Given	ABC is a △
Result	c = a + b
Abbr.	ext. ∠ of △

三角不等式 (Triangle Inequality)

Given	ABC is a △
Result	AB + BC > AC, BC + AC > AB, AB + AC > BC
Abbr.	N/A

等腰三角形 (Isosceles triangle)

Given	AB = AC
Result	∠B = ∠C
Abbr.	base ∠s, isos. △

Given	AB = AC and BD = DC
Result	∠BAD = ∠CAD and AD ⟂ BC
Abbr.	prop. of isos. △

Given	AB = AC and AD ⟂ BC
Result	BD = DC and ∠BAD = ∠CAD
Abbr.	prop. of isos. △

Given	AB = AC and ∠BAD = ∠CAD
Result	AD ⟂ BC and BD = DC
Abbr.	prop. of isos. △

Given	∠B = ∠C
Result	AB = AC
Abbr.	sides opp. equal ∠s

等邊三角形 (Equilateral triangle)

Given	AB = BC = AC
Result	∠A = ∠B = ∠C = 60°
Abbr.	prop. of equil. △

Given	∠A = ∠B = ∠C
Result	BC = AC = AB
Abbr.	prop. of equil. △

Congruent Triangle 全等三角形

SSS

Given	AB = XY and AC = XZ and BC = YZ
Result	△ABC ≅ △XYZ
Abbr.	SSS

SAS

Given	AB = XY and AC = XZ and ∠A = ∠X
Result	△ABC ≅ △XYZ
Abbr.	SAS

ASA

Given	AB = XY and ∠A = ∠X and ∠B = ∠Y
Result	△ABC ≅ △XYZ
Abbr.	ASA

AAS

Given	AB = XY and ∠A = ∠X and ∠C = ∠Z
Result	△ABC ≅ △XYZ
Abbr.	AAS

RHS

Given	AB = XY and AC = XZ and ∠C = ∠Z = 90°
Result	△ABC ≅ △XYZ
Abbr.	RHS

Corresponding Sides & Angles

Given	AB = XY and AC = XZ and △ABC ≅ △XYZ
Result	BC = YZ
Abbr.	corr. sides, ≅ △s

Given	AB = XY and AC = XZ and △ABC ≅ △XYZ
Result	BC = YZ
Abbr.	corr. sides, ≅ △s

Similar Triangle 相似三角形

AAA

Given	∠A = ∠X and ∠B = ∠Y and ∠C = ∠Z
Result	△ABC ~ △XYZ
Abbr.	AAA

3 sides proportional

Given	AB / XY = BC / YZ = CA / ZX
Result	△ABC ~ △XYZ
Abbr.	3 sides prop.

2 sides proportional and included angle equal

Given	AB / XY = AC / XZ and ∠A = ∠X
Result	△ABC ~ △XYZ
Abbr.	ratio of 2 sides, inc. ∠

Corresponding Sides & Angles

Given	△ABC ~ △XYZ
Result	AB / XY = BC / YZ = CA / ZX
Abbr.	corr. sides, ~△s

Given	△ABC ~ △XYZ
Result	∠A = ∠X and ∠B = ∠Y and ∠C = ∠Z
Abbr.	corr. ∠s, ~△s

Four Centre of Triangle 三角形四心

三角形內心 (Incentre)

Given	I is the incentre of △ABC
Result	I is the intersection of the angle bisectors, i.e. ∠BAX = ∠CAX, ∠ABY = ∠CBY, ∠BCZ = ∠ACZ
Abbr.	incentre of △

Given

I is the incentre of △ABC

Result

I is the intersection of the angle bisectors,

i.e. ∠BAX = ∠CAX, ∠ABY = ∠CBY, ∠BCZ = ∠ACZ

Abbr.

incentre of △

三角形外心 (Circumcentre)

Given	I is the circumcentre of △ABC
Result	I is the intersection of the perpendicular bisectors, i.e. IX ⟂ BC and BX = XC, IY ⟂ AC and AY = YC, IZ ⟂ AB and AZ = ZB
Abbr.	circumcentre of △

Given

I is the circumcentre of △ABC

Result

I is the intersection of the perpendicular bisectors,

i.e. IX ⟂ BC and BX = XC, IY ⟂ AC and AY = YC, IZ ⟂ AB and AZ = ZB

Abbr.

circumcentre of △

三角形形心 (Centroid)

Given	I is the centroid of △ABC
Result	I is the intersection of the medians, i.e. AZ = ZB, BX = XC, AY = YC and AI / IX = BI / IY = CI / IZ = 2 / 1
Abbr.	centroid of △

三角形垂心 (Orthocentre)

Given	I is the orthocentre of △ABC
Result	I is the intersection of the altitudes, i.e. AX ⟂ BC, BY ⟂ AC, CZ ⟂ AB
Abbr.	orthocentre of △

Parallelogram 平行四邊形

平行四邊形 (Properties of Parallelogram)

Given	ABCD is a parallelogram
Result	AB = DC and AD = BC
Abbr.	opp. sides of //gram

Given	ABCD is a parallelogram
Result	∠A = ∠C and ∠B = ∠D
Abbr.	opp. ∠s of //gram

Given	ABCD is a parallelogram and O is the intersection of diagonals
Result	AO = OC and BO = OD
Abbr.	diags. of //gram

證明平行四邊形 (Proof of Parallelogram)

Given	AB = DC and AD = BC
Result	ABCD is a parallelogram
Abbr.	opp. sides equal

Given	∠A = ∠C and ∠B = ∠D
Result	ABCD is a parallelogram
Abbr.	opp. ∠s equal

Given	AO = OC and BO = OD
Result	ABCD is a parallelogram
Abbr.	diags. bisect each other

Given	AD = BC and AD // BC
Result	ABCD is a parallelogram
Abbr.	opp. sides equal and //

Other Quadrilaterals 其他四邊形

長方形性質 (Properties of Rectangle)

Given	ABCD is a rectangle
Result	All properties of a parallelogram
Abbr.	prop. of rectangle

Given	ABCD is a rectangle
Result	All the interior angles are right angles
Abbr.	prop. of rectangle

Given	ABCD is a rectangle
Result	Diagonals are equal (AC = BD)
Abbr.	prop. of rectangle

Given	ABCD is a rectangle
Result	Diagonals bisect each other into four equal parts (AE = EC = BE = DE)
Abbr.	prop. of rectangle

菱形形性質 (Properties of Rhombus)

Given	ABCD is a rhombus
Result	All properties of a parallelogram
Abbr.	prop. of rhombus

Given	ABCD is a rhombus
Result	All sides are equal
Abbr.	prop. of rhombus

Given	ABCD is a rhombus
Result	Diagonals are perpendicular to each other (AC ⟂ BD)
Abbr.	prop. of rhombus

Given	ABCD is a rhombus
Result	Interior angles are bisected by the diagonals (a = b = c = d and e = f = g = h)
Abbr.	prop. of rhombus

正方形性質 (Properties of Square)

Given	ABCD is a square
Result	All properties of a rectangle
Abbr.	prop. of square

Given	ABCD is a square
Result	All sides are equal
Abbr.	prop. of square

Given	ABCD is a square
Result	Diagonals are perpendicular to each other (AC ⟂ BD)
Abbr.	prop. of square

Given	ABCD is a square
Result	Angles between each diagonal and a side are 45°
Abbr.	prop. of square

等腰梯形性質 (Properties Isosceles Trapezium)

Given	ABCD is an isos. trapezium
Result	AD // BC, AB = DC, AE = DF, AC = DB, BE = FC, AD = EF, ∠ABC = ∠DCB, ∠BAD = ∠CDA
Abbr.	prop. of isos. trapezium

鳶形性質 (Properties of Kite)

Given	ABCD is a kite
Result	AB = AD, BC = DC, ∠ABC = ∠ADC, a₁ = a₂, c₁ = c₂, b₁ = d₁, b₂ = d₂, AC ⟂ BD, BO = DO
Abbr.	prop. of a kite

Convex Polygon 凸多邊形

凸多邊形內角和 (Sum of interior angles of a convex polygon)

Given	a₁, a₂, a₃, … aₙ are the interior angles of an n-sided convex polygon
Result	a₁ + a₂ + a₃ + … + aₙ = (n − 2) × 180°
Abbr.	∠ sum of polygon

凸多邊形外角和 (Sum of exterior angles of a convex polygon)

Given	The exterior angles x₁, x₂, x₃, … xₙ of an n-sided convex polygon are taken in order
Result	x₁ + x₂ + x₃ + … + xₙ = 360°
Abbr.	sum of ext. ∠s of polygon

Basic Properties of Circles 圓的基本性質

圓心及弦 (Centre and Chord)

Given	AM = MB
Result	OM ⟂ AB
Abbr.	⟂ from centre to chord bisects chord

Given	AM = MB and
Result	OM ⟂ AB
Abbr.	line joining centre to mid-pt. of chord ⟂ chord

Given	CM ⟂ AB and AM = MB
Result	CM passes through the centre
Abbr.	⟂ bisector of chord passes through centre

圓心至弦的連線 (distance from centre to chord)

Given	AB = PQ
Result	OM = ON
Abbr.	equal chords equidistant from centre

Given	OM = ON
Result	AB = CD
Abbr.	chords equidistant from centre are equal

圓心角及圓周角 (angle at centre & angle at circumference)

Given	The angle at the centre and the angle at the circumference are subtended by the same arc AB
Result	∠AOB = 2∠ACB
Abbr.	∠ at centre twice ∠ at ⊙^ce

Given	AB is a diameter and C is a point on the circle
Result	∠ACB = 90°
Abbr.	∠ in semi-circle

Given	∠ACB = 90°
Result	AB is a diameter
Abbr.	converse of ∠ in semi-circle

Given	AB is a chord
Result	∠ACB = ∠ADB
Abbr.	∠s in the same segment

圓心角&弦&弧 (Angle at centre, Chord, Arc)

Given	∠AOB = ∠COD
Result	AB = CD
Abbr.	equal ∠s, equal chords

Given	AB = CD
Result	∠AOB = ∠COD
Abbr.	equal chords, equal ∠s

Given	∠AOB = ∠COD
Result	arc AB = arc CD
Abbr.	equal ∠s, equal arcs

Given	arc AB = arc CD
Result	∠AOB = ∠COD
Abbr.	equal arcs, equal ∠s

Given	arc AB = arc CD
Result	AB = CD
Abbr.	equal arcs, equal chords

Given	arc AB = arc CD
Result	AB = CD
Abbr.	equal arcs, equal chords

圓心角&圓周角&弧 (Angle at centre, Angle at circumference, Arc)

Given	∠AOB : ∠COD = m : n
Result	arc AB : arc CD = m : n
Abbr.	arcs prop. to ∠s at centre

Given	∠ADB : ∠BDC = m : n
Result	arc AB : arc BC = m : n
Abbr.	arcs prop. to ∠s at ⊙ce

圓内接四邊形 (Cyclic Quadrilateral)

Given	ABCD is a cyclic quadrilateral
Result	∠A + ∠C = 180° and ∠B + ∠D = 180°
Abbr.	opp. ∠s, cyclic quad.

Given	∠A + ∠D = 180° or ∠B + ∠C = 180°
Result	A, B, C and D are concyclic
Abbr.	opp. ∠s supp.

Given	One side of a cyclic quadrilateral is produced to form an exterior angle
Result	∠ADC = ∠EBC
Abbr.	ext. ∠, cyclic quad.

Given	DE and AB form a straight line and ∠ACD = ∠CBE
Result	A, B, C and D are concyclic
Abbr.	ext. ∠ = int. opp. ∠

Given	∠ACB = ∠ADB and both C and D are on the same side of AB
Result	A, B, C and D are concyclic
Abbr.	converse of ∠s in the same segment

半徑 vs 切線 (Radius vs Tangent)

Given	AB is the tangent to the circle at point T
Result	AB ⟂ OT
Abbr.	tangent ⟂ radius

Given	ATB ⟂ OT
Result	ATB is the tangent to the circle at T
Abbr.	converse of tangent ⟂ radius

切線性質 (Properties of Tangents)

Given	Two tangents drawn from an external point T meet the circle at points P and Q
Result	TP = TQ
Abbr.	tangent prop.

Given	Two tangents drawn from an external point T meet the circle at points P and Q
Result	∠TOP = ∠TOQ
Abbr.	tangent prop.

Given	Two tangents drawn from an external point T meet the circle at points P and Q
Result	∠OTP = ∠OTQ
Abbr.	tangent prop.

Given	PQ is the tangent to the circle at point A
Result	∠BCA = ∠BAP
Abbr.	∠ in alt. segment

Given	∠BCA = ∠BAP
Result	PQ is the tangent to the circle at A
Abbr.	converse of ∠ in alt. segment

雙圓 (Two Circles)

Given	Two circles touch each other either externally or internally
Result	O, A and O′ are collinear
Abbr.	prop. of two circles touching

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【Deductive Geometry 演繹幾何 Reasoning】直線│平行線│三角形│四邊形│多邊形│圓形【完整列表】

Deductive Geometry 演繹幾何 概覽

Angles related to lines & points 直線&點與角

同頂角 (Angles at a point)

Given

Result

Abbr.

直線上的鄰角 (Adjacent angles on a straight line)

Given

Result

Abbr.

對頂角 (Vertical opposite angles)

Given

Result

Abbr.

Parallel Lines 平行線

同位角 (Corresponding angles)

Given

Result

Abbr.

Given

Result

Abbr.

內錯角 (Alternate angles)

Given

Result

Abbr.

Given

Result

Abbr.

同旁內角 (Interior angles)

Given

Result

Abbr.

Given

Result

Abbr.

中點定理 (Mid-point Theorem)

Given

Result

Abbr.

截線定理 (Intercept Theorem)

Given

Result

Abbr.

Triangle 三角形

三角形內角和 (Angle sum of triangle)

Given

Result

Abbr.

三角形外角 (Exterior angle of triangle)

Given

Result

Abbr.

三角不等式 (Triangle Inequality)

Given

Result

Abbr.

等腰三角形 (Isosceles triangle)

Given

Result

Abbr.

Given

Result

Abbr.

Given

Result

Abbr.

Given

Result

Abbr.

Given

Result

Abbr.

等邊三角形 (Equilateral triangle)

Given

Result

Abbr.

Given

Result

Deductive Geometry 演繹幾何概覽