1. The document discusses the geometry sketching software The Geometer's Sketchpad (GSP) and provides examples of how to use it to explore properties of triangles, quadrilaterals, and trigonometric ratios. 2. GSP can be used to draw figures like triangle ABC and locate points like the midpoint X of side BC. It also allows investigation of theorems involving cross-ratios in configurations of four points like PQRS. 3. Users can explore angle properties and trigonometric ratios by drawing figures with points M, P, and Q and measuring angles and segment lengths, allowing relationships to be tested numerically.

2. C
AB
AB
C
AB
D
D

5. P
Q
P
AB
AB
Q

6. 1.
2.
3.
4.
5.
( SIMILARITY )
theorem
Menelaus'
(Ceva's theorem)

7. ∆ABC
BC, CA , AB
F
BD CE AF
DC EA FB
D,E,
1
A
F
B
E
C
D

8. S
A
F
∆ ABC
B
BD
A ,B,CCE AF
DC EA FB
F
1
S
D,E,
E
S
D
C

9. ABCD
u กับv และv sin θ
u v sin θ
1
u v sin θ
2
ABCD
ABD
AB u และ
AD v

10. 1.

11. 1.
P
CP
2.
CQ
CA
CB
AP
PB
AP
PB
AQ
BQ
AB
P,Q
∆CP
ABC CQ
C
Q AB
P,Q
CA AQ
CB QB
AB
C

12. 2.
A
ABC
B
C
BC
X AX

13. =
1.
BC
ABC
X
AX
2.
ˆ
A
B
P
ˆQ
A
AB BX
AX
C
AC XC
3.
AP
AQ
BX PX
AB AP
AC AQ
CX
APB
AQC QX
PX
AP AQ
QX
AQ QX
=
P,Q
BPX
=
AX
CQX

14. PQRS
PS
B
D
AB
QR
PR
C,D
AB
PQ
C
RS
PS
A ,
AB

15. A , PS
C
1.
2.
PQRS
QR
B
PS
AB
C,D
AD BS PQ
AB
DB SPQA
Menelaus BR QP
AC
AQB CB RQ PA
Menelaus
D
PQ
PR
RS
AB
1
1
CB RQ PA
CRPAC BR QP
1

16. 3.
PQB
4.
AD BS PQ
DB SP QA
AD CB
DB AC
SRA
PS
Menelaus BR QA 1
SBRQ AP
CB RQPA . PS BR QA
1.
AC BR QP
SBRQ AP
AC AD
CB BD
1
C,D
AB

17. AB
A
C
B

18. A
B
C
AB
1.
K
AK
L
L
2.
LN
3.
AB
A
AM
BL
N
D
CK
D
K
M
BK
AB

19. 1.
1.
2.
P
M
Q
ˆ
ˆ
sin A P sin A Q
M
M
ˆ
ˆ
sin A B sin B Q
M
M
l
AB
AB
MA , MB , MP
MQ
A , B , P และQ

20. P
Q
AP AQ
PB BQ
AB
พท AMP พท AMQ
..
..
พท PMB พท BMQ
..
..
1
ABC AB sin A C
BC ˆ
B
2
1
ˆ
AM sin A P
MP M
2
1
ˆ
PM sin P B
MB M
2
1
ˆ
AM sin A Q
MQ M
2
1
ˆ
BM sin B Q
MQ M
2
ˆ
ˆ
sin A P sin A Q
M
M
ˆ
ˆ
sin A B sin B Q
M
M
M
A'
A
P'
P
B'
B
Q'
l
Q

21. .
พื้นทีA MP
่' '
พื้นทีPMB
่' '
พื้นทีA MQ
่' '
พื้นทีBMQ
่' '
sin AMP sin AMQ
sin PMB sin BMQ
1
M
MAMPsin AMP MAsin AMP
' '
'
2
1
'
l
Q'
MPMBsin PMB MBsin PMB
' '
B'
P'
2
A'
1
MAMQsin AMP MAsin AMQ
' '
'
2
1
'
MBMQsin BMQ MBsin BMQ A
' '
P B
Q
2
'
'
พื้นทีA MP MAsin AMP MAsin AMQ พื้นทีA MQ
่' '
่' '
พื้นทีPMB MBsin PMB MBsin BMQ พื้นทีBMQ
่' '
'
'
่' '
A MP PMB A MQ ฺ' MQ
' ' , ' ' , ' ' และ B '
AP
' '
PB
' '
AQ
' '
BQ
' '
M
P
Q
AB

22. P,Q
AB
AB
AP,
AQ
P,Q
AB
AP AQ
PB BQ
AP PB AB AP
AQ BQ AQ AB
APAQ AB AQAB AP
(
)
(
)
2 1 1
AB AP AQ
AP, AB
AQ

23. 1.
1.1.
1.2.
1.3.
BC
A ,PS
X
ABC
AX
PQRS
QR
B
A
B
PS
PR
C
RS
AB

24. 2.
C
AB
A
C ,D
B
AB
D
AB

25. 3.
P
3.1
Q
AB
AB
ˆ
ˆ
sin A P sin A Q
M
MM
ˆ
ˆ
sin A B sin B Q
M
M
A3.2., P และQ
,B
MQ
3.3.
l
P,Q
MA , MB , MP
AB

26. 4.

27. •
•
•
•
•
•
•
.(
).
[
]
http://th.wikipedia.org/wiki/
25
2555.
[
]
http://th.wikipedia.org/wiki/
25
2555.
(2554).
MY MATHS.
.(2553)
.(2544). “
”
15
:
,
:
:
(Harmonic section).
2544

29. The Geometer’s Sketchpad
(GSP)

30. The Geometer’s
Sketchpad (GSP)
A
BC
X
AX
B
ABC
C

31. The Geometer’s Sketchpad
(GSP)
PQRS
B
D
PS
PR
RS
AB
C,D
A,PS
C
AB
QR
PS
AB

32. MQ
AB
AB
The Geometer’s Sketchpad (GSP)
P
Q
M
l
MA , MB , MP