Discussions rationnelles d`une courbe et d`un triangle

En

n
n a b c
½a2+b2=c2
ab = 2n
n
a b c
n n
u
v
uv a b c
½a2+b2=c2
ab = 2u
v
½(av)2+ (bv)2= (cv)2
(av) (bv) = 2uv
r s r
rs2
2
1
n
a b c
a2+b2=c2ab
2
x y z xn+yn=zn
n3

(3,4,5)
3×4
2= 6 (5,12,13)
30
¡m2−n2¢2+ (2mn)2=¡m2+n2¢2
a b a2+b2= 1 a6=−1
t
a=1−t2
1 + t2b=2t 1 + t2
(x, y)x2+y2= 1
1
O
A
M
A(−1,0) M
(a, b)
(AM)t
y=t(x+ 1)
a b
½x2+y2= 1
y=t(x+ 1)
x2+t2(x+ 1)2= 1 x=−1
22t2
1+t2
a=−2t2
1 + t2+ 1 = 1−t2 1 + t2

b
a b
t0 1
(a, b, c)¡a
c¢2+¡b c¢2= 1
t=m
n
a b
0 1
t(1−t2)
(1+t2)2t¡1−t2¢
0 1
21
2
31
3
2
34
2
4
1
2,1 3,2 3,1 4,3 4,1 5,2 5,3 5,4 5,1 6,5 6,1 7,2 7,3 7, . . .
6,6,30,15,21,30,210,15,5,210,330,21,70,210, . . .
n
n= 157
t=(526 771 095 761)2
(157 841)2×(4 947 203)2
1 2

1
a b c d
½a2+b2=c2
ab = 2d2
a b
c d
a b
m n
a=m2−n2;b= 2mn ;c=m2+n2
ab 2d22mn ¡m2−n2¢m
m2−n2x
m n21n m2−n2
mn ¡m2−n2¢
p q r
m=p2;n=q2;m2−n2=r2
r2=p4−q4=¡p2−q2¢¡p2+q2¢
1
s t s t s +t s −t
s=p2t=q2s+t=u2s−t=v2u2−v2=
(u+v) (u−v) = 2q2u v u +v u −v
u−v= 4a2u+v= 2b2
p2=v2+q2=b4+ 4a4
b2=s02−t02; 2a2= 2s0t0;p=s02+t02
s0t0s0t0
s0+t0s0−t0
s0< s
2