Euclidean Geometry
Theorem 1.
If
D
AOB is a straight line,
then
x
A
x + y = 180
(adj. s on st. line)
y
[ 直線上的鄰角]
O
B
Theorem 2, 3 (Converse of Theorem 1)
If
D
x + y = 180,
then
(adj. s supp.)
x y
O
A
AOB is a straight line
B
[直線上的鄰角逆定理]
或 [鄰角互補]
Corollary
w + x + y + z = 360
(s at a pt.)
w
x
[同頂角]
y z
Theorem 4.
If
A
D
x
O
two straight lines AOB, COD meet at O
then
y
x=y
(vert. opp. s)
C
[對頂角]
B
Theorem 5, 6, 7
a
A
then
B
If AB  CD
(1) a = b
c
(2) c = b
C
b
Theorem 8, 9, 10
d
D
(3) c + d = 180
(corr.s, AB//CD)
[同位角,AB//CD]
(alt.s, AB//CD)
[內錯角,AB//CD]
(int.s, AB//CD)
[同側內角,AB//CD]
(Converse of
(1)
Theorem 5, 6, 7)
If
then
(corr.s equal)
[同位角相等]
a
A
a=b
AB//CD
B
(2)
c
If
then
c=b
AB//CD
(alt.s equal)
[內錯角相等]
b
C
d
D
(3)
If
then
c + d = 180
AB//CD
(int.s supp.)
[同側內角互補]
1
Theorem 11.
A
If AB//CD and AB//EF
then CD//EF
B
C
D
(// to the same st. line)
[平行同一直線]
E
F
Theorem 12, 13
In ABC
(1) a + b + c = 180
A
( sum of )
[內角和]
(ext. of )
[的外角]
a
(2) d = a + b
b
c d
C
B
D
Theorem 14,15,16,17,18.
(Test for Congruent s)
(1)
In ABC, PQR
If AB = PQ, b = q, BC = QR
then ABC  PQR
A
(S.A.S)
a
(2)
b
c
B
(A.S.A.)
C
P
If b = q, BC = QR, c = r
then ABC  PQR
(3)
p
If
then
q
a = p, b = q, BC =QR
ABC  PQR
(A.A.S.)
r
Q
(4)
R
If AB = PQ, BC = QR, CA = RP
then ABC PQR
(S.S.S.)
P
A
(5)
If
then
Q
C
B
B = Q = 90, AC = PR, BC = QR
ABC  PQR
(R.H.S.)
R
Theorem 19, 20,21
(Tests for Similar Triangles)
(1)
A
In ABC, PQR
If a = p and b = q and c = r
then ABC  PQR
a
(A.A.A)
(2)
B
b
If
then
P
p
a = p and
AB AC

PQ PR
ABC  PQR
(ratio of 2 sides, inc.)
[兩對邊成比例，夾角相等]
c
Q q
C
(3)
If
then
r
R
AB AC BC


PQ PR QR
ABC  PQR
(3 sides proportional)
[對應邊成比例]
2
Theorem 22, 23.
(1) The sum of the interior angles of a convex polygon with n sides is
( sum of
(2)
(n-2)x180
polygon) [多邊形內角和]
If the sides of a convex polygon are produced in order, the sum of the exterior angles
so formed is 360
(sum of ext.s of polygon)
[多邊形外角和]
Theorem 24
A
If ABC is isosceles such that AB = AC
then B = C
(base s, isos.)
[等腰的底角]
B
Theorem 25
C
(Converse of Theorem 24)
A
If B = C
then AC = AB
(sides opp. equal s)
[等角的對邊]
B
C
Theorem 26
A
If AB = BC = CA
then A = B = C = 60
(Property of equilateral )
[等邊性質]
B
C
Theorem 27, 28,29, 30.
A
B
a
b
O
c
C
then
If ABDC is a parallelogram
(1) AB = CD; AC = BD
(opp. sides, //gram) [平行四邊形對邊]
(2) a = d, c = b
(opp. s, //gram) [平行四邊形對角]
d
D
(3) AO = OD; CO = OB
(diagonals //gram)
[平行四邊形對角線]
(4) area of ABC = area of DCB;
area of ADC = area of DAB
(diagonals bisect area of //gram)
[平行四邊形被對角線平分]
3
Theorem 31,32,33 34.
In quadrilateral ABCD,
if AB = DC and AD = BC
then ABCD is a parallelogram
(1)
(Tests for Parallelograms)
A
B
a
b
O
d
(opp. sides equal)
(2)
if a = c and b = d
then ABCD is a parallelogram
(3)
if AO = OC and BO = OD
then ABCD is a parallelogram
(opp.s equal)
c
D
C
Theorem 35.
[對角線互相平分]
if AB = DC and AB // DC
then ABCD is a parallelogram
(opp. sides equal and //) [對邊平行且相等]
If ABDC is a square,
then (1) AD = BC
(2) AD  BC
(3) AD bisects BAC and BDC;
BC bisects ABD and ACD
(4) possess all properties of a parallelogram
(property of square) [正方形性質]
B
O
C
[對角相等]
(diagonals bisect each other)
(4)
A
[對邊相等]
D
Theorem 36.
A
If ABCD is a rectangle,
then (1) AC = BD
(2) possess all properties of a parallelogram
(property of rectangle) [矩形性質]
B
O
D
C
Theorem 37.
If ABCD is a rhombus,
then (1) AC  BD
(2) AC bisects BAD, BCD;
BD bisects ABC, ADC
(3) possess all properties of a parallelogram
(property of rhombus) [菱形性質]
A
D
B
C
Theorem 38.
(Mid-point theorem)
In ABC,
If
D, E are mid-points of AB, AC respectively
then (1) DE // BC
(2) DE = BC/2
(Mid-pt. theorem) [中點定理]
A
E
D
B
C
Theorem 39.
C
E
G
(Equal intercept theorem)
A
B
D
If
AB // CD // EF // GH and
AC = CE = EG
then BD = DF = FH
(Equal intercept theorem) [等截距定理]
or (Intercept theorem) [截線定理]
F
H
4
Theorem 40.
(Intercept theorem)
A
E
In ABC,
If D is a mid-point of AB; DE // BC
then AE = EC
D
(Intercept theorem)
[截線定理]
B
Theorem 41.
C
(Theorem of equal ratio)
A
D, E divide AB, AC
internally
D
E
B
C
A
D, E divide AB, AC
externally
C
B
If DE // BC
then
AD AE

DB EC
(Equal ratios theorem)
D
[等比定理]
E
E
D
D, E divide AB, AC
externally
A
B
C
Theorem 42.
(Converse of Theorem 42)
A
D, E divide AB, AC
internally
D
AD AE

DB EC
If
E
then
B
C
(Converse of equal ratios theorem)
A
D, E divide AB, AC
externally
C
B
D
DE // BC
[等比定理之逆定理]
E
E
D
A
B
D, E divide AB, AC
externally
C
5
Theorem 43.
(Pythagoras' theorem)
In
A
ABC
B = 90
then AB2  BC2  AC2
(Pythagoras' theorem)
[畢氏定理]
B
C
(Converse of theorem 43)
Theorem 44.
A
In
ABC
AB2  BC2  AC2
then B = 90
(Converse of Pythagoras' theorem)
[畢氏定理之逆定理]
C
B
Theorem 45.
(Perpendicular bisector theorem)
H
If HK is the perpendicular bisector of AB
P in a point on HK
then PA = PB
P
( bisector theorem)
A
B
[中垂線定理]
K
Theorem 46.
(Converse of Theorem 46)
H
If HK is the perpendicular bisector of AB
PA = PB
then P is a point on HK
P
A
B
(Converse of  bisector theorem)
[中垂線定理之逆定理]
K
Theorem 47.
(Angle bisector theorem)
A
E
F
P
B
C
D
(Converse of Theorem 47)
A
Theorem 48.
E
F
P
B
C
If AD is the angle bisector of BAC
P is a point on AD
PE is the perpendicular distance of P from AB
PF is the perpendicular distance of P from AC
then PE = PF
( bisector theorem)
[角平分線定理]
If AD is the angle bisector of BAC
PE is the perpendicular distance of P from AB
PF is the perpendicular distance of P from AC
PE = PF
then P is a point on AD
(Converse of  bisector theorem)
[角平分線定理之逆定理]
D
6
Theorem 49.
In
A
c
ABC,
a+b>c
b+c>a
c+a>b
b
(triangular inequality)
B
[三角形不等式]
C
a
In ABC,
(1) If a > b , then A > B .
Theorem 50.
A
(greater side, greater opp. )
c
b
(2)
[長邊對應大角]
If A > B, then a > b .
(greater , greater opp. side)
[大角對應長邊]
B
C
a
Theorem 51a.
(Centroid theorem)
ABC,
AD, CE, BF are the medians
then (1) AD, CE and BF meet at a point, G.
(G is the centroid of the triangle)
(2) AG:GD = BG:GF = CG:GE = 2:1;
In
A
F
E
G
(Centroid theorem)
B
D
C
(Circum-centre theorem)
Theorem 51b.
[重心定理]
ABC,
DE, GF, KH are the perpendicular bisectors
of the sides AB, AC and BC respectively
then DE, GF and KH meet at a point, O.
(O is the circumcentre of the triangle)
In
A
G
H
D
O
(Circum-center theorem)
E
C
F
[外接圓心定理]
K
B
Theorem 51c.
(In-center theorem)
F
O
E
ABC,
AD, BF, CE are the angle bisectors of the
angles of the triangle
then AD, BF, CE meet at a point, O.
(O is the in-centre of the triangle)
In
A
D
C
(In-centre theorem)
[內切圓心定理]
B
Theorem 51d.
(Ex-centre theorem)
A
ABC,
AD is the angle bisector of a interior angle,
BE and CF are the angle bisectors of the
exterior angles of the other angles
then AD, BE, CF meet at a point, O.
(O is the ex-centre of the triangle)
C
In
B
E
F
D
(Ex-centre theorem)
[旁切圓心定理]
7
Theorem 51e.
(Orthocentre theorem)
A
E
then
O
B
ABC,
AD, BF, CE are the altitudes
AD, BF, CE meet at a point, O.
(O is the orthocenter of a triangle)
(Orthocentre theorem) [垂心定理]
In
F
D
C
Theorem 52,53.
A
D
O
If AB CD then arcAB arcCD.
(equal chords, equal arcs) [等弦,等弧]
Conversely
If arcAB arcCD then AB CD.
(equal arcs, equal chords) [等弧,等弦]
B
C
Theorem 54,55 (corollary of
theorem 52, 53)
A
C
B
D
Two equal circles.
If AB CD then arcAB arcCD.
(equal chords, equal arcs) [等弦,等弧]
Conversely
If arcAB arcCD then AB CD.
(equal arcs, equal chords) [等弧,等弦]
Theorem 56, 57,58,59.
If arcAB arcCD (or AB CD) then
A
(equal arcs, equal s) [等弧,等角]
D
(or equal chords, equal s
O
m
m  n.
等弦,等角)
Conversely
n
If
m  n then arcAB arcCD (or AB CD).
(equal s, equal arcs) [等角,等弧]
B
(or equal s, equal chords 等角,等弦)
C
Corollary 60,61,62, 63.
Two equal circles.
If arcAB arcCD (or AB CD) then
m  n.
(equal arcs, equal s) [等弧,等角]
A
C
m O
n P
(or equal chords, equal s
Conversely
If
m  n then arcAB arcCD (or AB CD).
(equal s, equal arcs) [等角,等弧]
D
B
等弦,等角)
(or equal s, equal chords 等角,等弦)
Theorem 64.
arcAB : arcBC  m : n
(arcs prop. to s at centre)
O
m n
A
[弧與圓心角成比例]
C
B
8
Corollary 65.
Two equal circles.
A
C
m O
P
n
arcAB : arcCD  m : n
(arcs prop. to s at centre)
[弧與圓心角成比例]
D
B
Theorem 66.
m
arcAB : arcBC  m : n
n
(arcs prop. to s at circumference)
O
A
[弧與圓周角成比例]
C
B
Corollary 67.
Two equal circles.
arcAB : arcCD  m : n
n
A m
(arcs prop. to s at circumference)
C
[弧與圓周角成比例]
D
B
Theorem 68.
If
then AN  NB.
(line from centre  chord bisects chord)
O
A
[圓心至弦之垂線平分該弦]
B
N
Theorem 69. (Converse of Theorem 68)
AN  NB
then ON  AB.
If
(line joining centre to mid-pt. of chord perp.
to chord)
O
A
ON AB
[弦的中點與圓心聯線  該弦]
B
N
Theorem 70.
B
AB  CD
then OM  ON .
If
M
A
(equal chords, equidistant from centre)
O
[等弦與圓心等距]
D
N
C
Theorem 71. (Converse of Theorem 70)
OM  ON
then AB  CD.
If
B
M
A
[與圓心等距的弦等長]
O
C
(chords equidistant from centre are equal)
N
D
9
Theorem 72.
P
AOB  2APB
(  at centre twice  at circumference)
O
[圓心角兩倍於圓周角]
O
B
P
A
A
P
A
O
B
B
Theorem 73.
C
A
AB is a diameter,
then ACB  90o
If
B
O
(  in semi-circle)
[半圓上的圓周角]
Theorem 74.
P
AB is a chord,
then APB  AQB.
If
Q
(s
in the same segment)
[同弓形內的圓周角]
A
B
Theorem 75.
S
If
R
P
PQRS is a cyclic quadrilateral,
then P  R  180o ( or S  Q  180o)
(opp.  s, cyclic quad.)
[圓內接四邊形對角]
Q
Theorem 76.
S
If
R
PQRS is a cyclic quadrilateral,
then S  RQK
(ext.  s, cyclic quad.)
[圓內接四邊形外角]
P
K
Q
Theorem 77. (Converse of Theorem 74)
P
If
Q
APB  AQB
then A, B, Q, P are concyclic.
(Converse of  s in the same segment)
[同弓形內的圓周角的逆定理]
A
B
10
Theorem 78. (Converse of Theorem 75)
S
P  R  180o ( or S  Q  180o)
If
R
then P, Q, R, S are concyclic.
(opp.  s supp)
[對角互補]
P
Q
Theorem 79. (Converse of Theorem 76)
S
R
P
If S  RQK
then P, Q, R, S are concyclic.
(ext.  = int. opp.  )
[外角=內對角]
K
Q
Theorem 80.
If PQ is a tangent to the circle ,
then PQ OT .
(tangent  radius)
[切線  半徑]
O
T
P
Q
Theorem 81. (Converse of Theorem 80)
If
then
O
T
P
PQ OT ,
PQ is a tangent to the circle.
(Converse of tangent 
[切線  半徑的逆定理]
radius)
Q
Theorem 82.
If TP, TQ are two tangents to the circle,
then (1) TP=TQ
(2) TOP  TOQ
(3) OTP  OTQ
P
O
T
(tangent properties)
[切線性質]
Q
Theorem 83.
C
B
If
then
P
A
PAQ is a tangent to the circle at A,
CAQ  CBA (or BAP  BCA)
(  in alt. segment)
[交錯弓形的圓周角]
Q
11
Theorem 84. (Converse of Theorem 83)
C
B
If
then
CAQ  CBA (or BAP  BCA),
PAQ is a tangent to the circle at A.
(Converse of  in alt. segment)
P
A
[交錯弓形的圓周角的逆定理]
Q
12