Download 【Plane Geometry 平面幾何】Revision Notes 溫習筆記

【Plane Geometry 平面幾何】Revision Notes 溫習筆記
Angles and Parallel Lines
The sum of all the angles If two straight lines
intersect, the vertically
at a point is 360°.
opposite angles are
equal.
b
a
c
a
The sum of all the
adjacent angles on a
straight line is 180°.
a
角與平行線
b
b
a + b = 180o
[adj. Ðs on st. line]
[直線上的鄰角]
a + b + c = 360o
[Ðs at a pt.]
[同頂角]
a=b
If AB//CD, then a = b .
If AB//CD, then a = b .
If AB//CD, then
a + b = 180°.
A
C
a
b
B
D
[corr. Ðs, AB//CD]
[同位角， AB//CD]
A
b
C
a
[vert. opp. Ðs]
[對頂角]
B
D
[alt. Ðs, AB//CD]
[錯角， AB//CD]
A
a
b
C
B
D
[int. Ðs, AB//CD]
[同旁內角， AB//CD]
The converse of the above 3 theorems can be used as a test for parallel lines.
[corr. Ðs equal; alt. Ðs equal; int. Ðs, supp.][同位角相等、錯角相等、同旁內角互補]
Angles of a Triangle and Convex Polygon 三角形的角及凸多邊形的角
The sum of interior angles of a triangle The exterior angle of a triangle is
equal to the sum of interior opposite
is 180°.
angles.
a
a
c
c1
b
b
a + b + c = 180o
[Ð sum of D]
[D內角和]
The sum of the interior angles of a
convex polygon with n sides is
(n - 2) ´ 180°.
[Ð sum of polygon]
[多邊形內角和]
a + b = c1
[ext. Ð of D]
[D外角]
The sum of exterior angles of a convex
polygon with n sides is 360°.
[ext. Ð of polygon]
[多邊形外角和]
[r-note-1.doc]
p.1
Conditions for Congruent Triangles
證明全等三角形的條件
[S.S.S.]
[S.A.S]
[A.S.A]
[R.H.S]
If two Ds are congruent, their corresponding angles and sides are equal.
若兩三角形全等，則所對應的角及邊相等。
Conditions for Similar Triangles
a
b
7
5
c
a
6
b
c
證明相似三角形的條件
5
6
14
10
10
12
12
[3 sides proportional]
[三邊成比例]
[equiangular]
[等角]
[ratio of 2 sides, inc. Ð]
[二邊成比例且夾角相等]
If two Ds are similar, their corresponding angles are equal, and their
corresponding sides are proportional.
若兩三角形相似，則所對應的角相等及對應邊成比例。
Isosceles Triangle
AB = AC , then b = c . If
If
b = c , then AB = AC . If AB = AC, and BM =
MC
A
then AM ^ BC and d = e .
A
A
B
b
c
C
B
[base Ðs, isos. D]
[等腰D底角]
等腰三角形
b
c
M
B[properties
of isos. D]C
[等腰D性質]
[sides. opp. equal Ðs]
[等角對邊相等]
Equilateral Triangle
If AB = BC = CA, then a = b = c
A
= 60 o .
B
d e
C
等邊三角形
If a = b = c = 60 o , then AB = BC =
A
CA .
C
[properties of equil. D]
B
C
[等邊D性質]
[r-note-1.doc]
p.2
Parallelograms
平行四邊形
If ABCD is a parallelogram , then
ÐA = ÐC , ÐB = ÐD .
AB = DC , AD = BC .
AO = OC , BO = OD .
A
A
D
D
A
D
O
B
B
C
C
B
C
[opp. sides of // gram]
[opp. Ðs of // gram]
[diagonals of // gram]
[//四邊形對邊]
[//四邊形對角]
[//四邊形對角線]
The converse of each of the 3 theorems can be used as a test for parallelogram.
J a parallelogram is a quadrilateral with parallel opposite sides.
J a rectangle is a parallelogram with right interior angles.
J a rhombus(菱形) is a parallelogram with equal adjacent sides.
J a square is a rectangle with equal adjacent sides.
Mid-Point and Intercept Theorems
If AE = EB, AF = FC ,
then
EF // BC and
1
EF =
BC
2
A
中點及截線定理
If AE = EB and EF // BC , If AB // CD // EF and
then AF = FC .
AC = CE , then BD = DF .
A
B
A
E
F
D
C
F
E
C
B
C
B
[mid-point theorem]
[中點定理]
[intercept theorem]
[截線定理]
Equal Ratio Theorem & Its Converse
In DABC, if EF // BC then
A E AF
=
.
E B FC
A
E
B
[intercept theorem]
[截線定理]
等比定理及逆定理
If EF divides AB and AC such that
A E AF
=
then EF // BC .
E B FC
A
3
E
F
C
[equal ratio theorem]
[等比定理]
F
E
6
4
F
8
C
B
[converse of equal ratio theorem]
[等比逆定理]
[r-note-1.doc]
p.3
Pythagoras’ Theorem & Its Converse
If ÐC = 90°, then c 2 = a 2 + b 2 .
If c 2 = a 2 + b 2 , then ÐC = 90°.
c
a
畢氏定理及逆定理
c
a
b
b
[Pythagoras’ theorem]
[畢氏定理]
[converse of Pythagoras’ theorem]
[畢氏定理的逆定理]
Perpendicular Bisector
If EF is the perpendicular bisector of
AB and P is any point on EF, then P is
equidistant from A and B.
E
P
A
垂直平分線
If P is equidistant from two given
points A and B, then P lies on the
perpendicular bisector of AB .
E
P
A
B
B
F
[converse of ^ bisector theorem]
[垂直平分線定理的逆定理]
F
[^ bisector theorem]
[垂直平分線定理]
Angle Bisector
角平分線
If P is equidistant from OA and OB,
then P is a point on the angle bisector
of ÐAOB .
A
If ON is the bisector of ÐAOB and if P
is any point on ON, then P is
equidistant from OA and OB .
A
N
N
P
P
O
B
O
[converse of Ð bisector theorem]
[角平分線定理的逆定理]
B
[Ð bisector theorem]
[角平分線定理]
Chords of a Circle
If ON ^ AB , then AN =
NB .
圓的弦
If AN = NB , then ON ^
AB .
If AN = NB and PN ^AB，
then O is on PN .
P
O
O
A
N
B
[^ from centre bisects
chord]
[圓心至弦的垂線平分弦]
A
N
B
A
N
B
[line joining centre to
[^ bisector of chord passes
mid-pt. of chord]
through centre]
[圓心至弦中點的連線^弦] [弦的^平分線通過圓心]
[r-note-1.doc]
p.4
If AB = CD , then OM = ON .
If OM = ON , then AB = CD .
C
C
M
O
A
M
O
D
B
N
A
D
B
N
[equal chords, equidistant from centre] [chords equidistant from centre are equal]
[等弦與圓心等距]
[與圓心等距的弦等長]
Angles in a Circle
圓上的角
If P lies on the circumference of a circle, then
x = 2y .
x=y.
P
P
y
y
O
x
A
x
B
B
A
[Ðs in the same segment]
[同弓形內的圓周角]
Angles, Arcs and Chords
CD
=
x
.
y
x O
B
y
B
[Ð in semi-circle]
[半圓上的圓周角]
角、弧、弦
A
C
A
O
In the same circle (or equal circles),
equal angles Û equal chords Û equal
C
arcs
In the same circle (or equal circles),
AB
P
x
Q
A
[Ð at centre twice Ð at
circumference]
[圓心角兩倍於圓周角]
)
)
If AOB is a diameter,
then x = 90°.
O
D
A
x
C
D
D
D
y
C
B
A
B
B
[equal angles , equal chords]
[Ðs at centre are proportional to arcs]
[equal angles , equal arcs] , etc
etc
[圓心角和圓周角與所對的弧成比例]
Note : Ðs at centre / circumference are NOT proportional to chords
[r-note-1.doc]
p.5
Cyclic Quadrilateral
圓內接四邊形
If ABCD is a cyclic quadrilateral, then
x + y = 180°.
x=y.
D
D
y
y
A
A
C
C
x
x
B
[opp. Ðs, cyclic quad.]
[圓內接四邊形對角]
B
[ext. Ðs, cyclic quad.]
[圓內接四邊形外角]
Tests for Cyclic Quadrilateral
If x = y then A,B,C,
D are concyclic .
D
y
If x + y = 180° then
A,B,C, D are concyclic .
D
C
A
y
x
If x = y the A,B,C, D
are concyclic .
D
y
A
C
B
A
四點共圓的驗證
C
x
[equal Ðs in the same
segment]
[同弓形內的圓周角的逆定
理]
x
B
[opp. Ðs supp.]
[對角互補]
B
[ext. Ð = int. opp. Ð]
[外角等於內對角]
Tangent Properties
切線的性質
If AB is a tangent to the If AB and AC are
If AB is tangent to the
circles with centre at O,
tangents to the circle at circle at A and AB is a
B and C respectively,
at T, then AB^OT
chord, then x = y .
then
AB = AC ; a = b and
Q
x=y.
C
O
y
O y
x
A
P
B
a
b
A
A
x
P
B
B
[tangent perp. to radius]
[tangent properties]
[Ð in alt. segment]
[切線垂直半徑]
[切線性質]
[交錯弓形的圓周角]
The converse of each of the 3 theorems can be used as a test for a tangent.
[r-note-1.doc]
p.6
incertre , circumcentre , orthocentre and centroid in a triangle
Circumcentre 外心
Incertre 內心
intersection of the perpendicular bisector,
intersection of the angle bisector, and the
and the centre of the circumscribed circle.
centre of the inscribed circle.
c c
a
a
a
Orthocentre 垂心
intersection of the altitude
c
c
a
b
b
b
b
Centroid 形心
intersection of the median
a
c
a
c
b
b
[r-note-1.doc]
p.7