DSE 數學公式 (Maths Formula): Deductive Geometry 演繹幾何定理公式

Deductive Geometry

Angles	Properties
	a + b = 180° adj. ∠s on st. line
	a + b + c = 360° ∠s at a pt.
	a = b vert. opp. ∠s
	a + b + c = 180° ∠ sum of Δ
	a + b = c_1 ext. ∠ of Δ *快捷版，可不記

	Properties	Prove
Parallel lines ***見到F/Z/C shape先好用
	If AB // CD, then a = b corr. ∠s, AB // CD	If a = b, then AB // CD corr. ∠s equal
	If AB //CD, then a = b alt. ∠s, AB // CD	If a = b, then AB //CD alt. ∠s equal
	If AB // CD, then a + b = 180° int. ∠s, AB // CD	If a + b = 180°, then AB // CD int. ∠s supp.

Angles of convex polygon
	a_1 + a_2 + a_3 + … + a_n = (n − 2) × 180° ∠ sum of polygon → Regular 正多邊形 Each interior angle = \frac{(n − 2) × 180°}{n}
	x_1 + x_2 + x_3 + … + x_n = 360° sum of ext. ∠s of polygon → Regular 正多邊形 Each exterior angle = \frac{360º}{n}

/ Δs
Congruent Triangles (P.S. congruent Δ 係 similar Δ其中一種）		Similar Triangles
If ΔABC ≅ΔXYZ, then ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z (corr. ∠s, ≅Δs) AB = XY, BC = YZ, CA = ZX (corr. sides, ≅Δs)   If ΔABC ~ ΔXYZ, then ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z (corr. ∠s, ≅Δs) \frac{AB}{XY} = \frac{BC}{YZ} = \frac{CA}{ZX} (corr. sides, ~Δs)

Properties of quadrilateral
	Properties	Prove
	∠P + ∠Q = 180° ∠R + ∠S = 180º Prop. of trapezium
	90º + all prop. of //gram
	AD = BC & AB = DC ∠A = ∠C & ∠B = ∠D AO = CO & BO = DO AD = BC & AD // BC prop. of //gram	opp. sides equal opp. ∠s equal diags. bisect each other 2 sides equal and //
	diagonals are ⊥ 4 sides equal Interiors are bisected by diagonals + all prop. of //gram Properties of rhombus

→ Right-angled triangle

🌟見到直角唔係畢氏定理就係sin/cos/tan

sin θ = \frac{opp.對}{hyp.斜} = \frac{a}{c}

cos θ = \frac{adj.鄰}{hyp.斜} = \frac{b}{c}

tan θ = \frac{opp.對}{adj.鄰} = \frac{a}{b}

a^2 + b^2 = c^2 (pyth. Them.)

If a^2 + b^2 = c^2, then ∠C = 90º (Converse of Pyth. Thm.)

→ Isosceles triangle

If AB = AC, then ∠ABC = ∠ACB (base ∠s, isos. Δ)

If ∠ABC = ∠ACB, then AB = AC (sides opp. equal ∠s)

If ΔABC is an isos. Δ,

then AD ⊥ BC & BD = DC & ∠BAD = ∠CAD (prop. of isos. Δ)

→ Equilateral Triangle

If AB = BC = CA, then ∠A, ∠B, ∠C = 60º, vice versa (prop. of equil. Δ)

→ Mid-point theorem and Intercept Theorem

If AE = EB and AF = FC, then EF // BC & EF = \frac{1}{2}BC (Mid-pt. theorem)

If AE = EB and EF // BC, then AF = FC (intercept theorem)

If AC = CE and AB // CD // EF, then BD = DF (intercept theorem)

→ Triangle inequality

a + b > c

b + c > a

c + a > b

→ Sine formula (2邊2角）：

\frac{a}{sinA} = \frac{b}{simB} = \frac{c}{sinC}

→ Consine formula (3邊1角）：

c^2 = a^2 + b^2 − 2abcosC

→ Center formula

→ Symmetry

Axis of symmetry of n-sided regular polygon = n

Order of rotational symmetry of n-sided regular polygon = n

Number of diagonals of n-sided polygon = \frac{n(n − 3)}{2}

演繹幾何定理

Angles	特性
	a + b = 180° 直線上的鄰角
	a + b + c = 360° 同頂角
	a = b 對頂角
	a + b + c = 180º Δ內角和
	a + b = c_1 Δ外角 *快捷版，可不記

	Properties	Prove
平行線 ***見到F/Z/C shape先好用
	If AB // CD, then a = b 同位角，AB // CD	If a = b, then AB//CD 同位角相等
	If AB // CD, then a = b 錯角，AB // CD	If a = b, then AB // CD 錯角相等
	If AB // CD, then a + b = 180° 同旁內角，AB //CD	If a + b = 180º, then AB // CD 同旁內角互補

凸多邊形的角
	a_1 + a_2 + a_3 + … + a_n = (n − 2) × 180° 多邊形的內角和 → Regular 正多邊形 每隻內角 = \frac{(n − 2) × 180°}{n}
	x_1 + x_2 + x_3 + … + x_n = 360° 多邊形的外角和 → Regular 正多邊形 每隻外角 = \frac{360°}{n}

/ Δs
全等三角形 （P.S. 全等三角形 係 相似三角形 其中一種）		相似三角形
If ΔABC ≅ ΔXYZ, then ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z (全等三角形對應角） \frac{AB}{XY} = \frac{BC}{YZ} = \frac{CA}{ZX} （相似三角形對應邊）

四邊形的特性
	特性	證明
	∠P + ∠Q = 180° ∠R + ∠S = 180º 梯形性質
	90º + 所有平行四邊形性質
	AD = BC & AB = DC ∠A = ∠C & ∠B = ∠D AO = CO & BO = DO AD = BC & AD // BC 平行四邊形性質	對邊相等 對角相等 對角線互相平分 一組對邊相等且平行
	對角線互相垂直 4 邊相等 對角線平分頂角 + 所有平行四邊形性質 菱形性質

→ 直角三角形

🌟 見到直角唔係畢氏定理就係 sin/cos/tan

sin θ = \frac{對}{斜} = \frac{a}{c}

cos θ = \frac{鄰}{斜} = \frac{b}{c}

tan θ = \frac{對}{鄰} = \frac{a}{b}

a^2 + b^2 = c^2

If a^2 + b^2 = c^2, then ∠C = 90º（畢氏定理逆定理）

→ 等腰三角形

If AB = AC, then ∠ABC = ∠ACB（等腰Δ底角）

If ∠ABC = ∠ACB, then AB = AC（等角對邊相等）

If ΔABC is an isos. Δ, then AD ⊥ BC & BD = DC & ∠BAD = ∠CAD（等腰Δ性質）

→ 等邊三角形

If AB = BC = CA, then ∠A, ∠B, ∠C = 60º, vice versa（等邊Δ性質）

→ 中點定理及截線定理

If AE = EB and AF = FC, then EF // BC & EF = \frac{1}{2}BC（中點定理）

If AE = EB and EF // BC, then AF = FC（截線定理）

If AC = CE and AB // CD // EF, then BD = DF（截線定理）

→ 三角不等式

a + b > c

b + c > a

c + a > b

→ 正弦公式（2邊2角）

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

→ 餘弦公式（3邊1角）

c^2 = a^2 + b^2 − 2ab cos C

→ 三角形的中心

→ 對稱

正n邊形的對稱數 = n

正n邊形的旋轉折式數目 = n

正n邊形的對角線 = \frac{n(n−3)}{2}