The configuration consists of the unit circle centered at the
origin and a straight line passing through the point (-1,0) which lies on the
circle. Unless the line is tangent to the circle, the two have a second common
point. In order to find this point, we have to solve simultaneously two
equations: the quadratic equation of the circle x2 + y2 = 1 and the linear
equation of the line. By eliminating either x or y from the latter, and
substituting the result into the former, we get a quadratic equation in one
variable with integer coefficients. One solution of this equation is immediate -
it is related to the point (-1,0), and is rational. Therefore, the second
solution of the equation is also rational and gives either x- or y-coordinate of
the second point of intersection. (3,4,5), (5,12,13), (6,8,10), (7,24,25),
(8,15,17), (9,12,15), (9,40,41), (10,24,26), (11,60,61), (12,16,20), (12,35,37),
(13,84,85), (14,48,50), (15,20,25), (15,36,39), (15,112,113), (16,30,34),
(16,63,65), (17,144,145), (18,24,30), (18,80,82), (19,180,181), (20,21,29),
(20,48,52), (20,99,101), (21,28,35), (21,72,75), (21,220,221), (22,120,122),
(23,264,265), (24,32,40), (24,45,51), (24,70,74), (24,143,145), (25,60,65),
(25,312,313), (26,168,170), (27,36,45), (27,120,123), (27,364,365), (28,45,53),
(28,96,100), (28,195,197), (29,420,421), (30,40,50), (30,72,78), (30,224,226),
(31,480,481), (32,60,68), (32,126,130), (32,255,257), (33,44,55), (33,56,65),
(33,180,183),
(33,544,545), (34,288,290), (35,84,91), (35,120,125),
(35,612,613), (36,48,60), (36,77,85), (36,105,111), (36,160,164), (36,323,325),
(37,684,685), (38,360,362), (39,52,65), (39,80,89), (39,252,255), (39,760,761),
(40,42,58), (40,75,85), (40,96,104), (40,198,202), (40,399,401), (41,840,841),
(42,56,70), (42,144,150), (42,440,442), (43,924,925), (44,117,125),
(44,240,244), (44,483,485), (45,60,75), (45,108,117), (45,200,205),
(45,336,339), (46,528,530), (48,55,73), (48,64,80), (48,90,102), (48,140,148),
(48,189,195), (48,286,290), (48,575,577), (49,168,175), (50,120,130),
(50,624,626), (51,68,85), (51,140,149), (51,432,435), (52,165,173),
(52,336,340), (52,675,677), (54,72,90), (54,240,246), (54,728,730),
(55,132,143), (55,300,305), (56,90,106), (56,105,119), (56,192,200),
(56,390,394), (56,783,785), (57,76,95), (57,176,185), (57,540,543),
(58,840,842), (60,63,87), (60,80,100), (60,91,109), (60,144,156), (60,175,185),
(60,221,229), (60,297,303), (60,448,452), (60,899,901), (62,960,962),
(63,84,105), (63,216,225), (63,280,287), (63,660,663), (64,120,136),
(64,252,260), (64,510,514), (65,72,97), (65,156,169), (65,420,425), (66,88,110),
(66,112,130), (66,360,366), (68,285,293), (68,576,580), (69,92,115),
(69,260,269), (69,792,795), (70,168,182), (70,240,250), (72,96,120),
(72,135,153), (72,154,170), (72,210,222), (72,320,328), (72,429,435),
(72,646,650), (75,100,125), (75,180,195), (75,308,317), (75,560,565),
(75,936,939), (76,357,365), (76,720,724), (77,264,275), (77,420,427),
(78,104,130), (78,160,178), (78,504,510), (80,84,116), (80,150,170),
(80,192,208), (80,315,325), (80,396,404), (80,798,802), (81,108,135),
(81,360,369), (84,112,140), (84,135,159), (84,187,205), (84,245,259),
(84,288,300), (84,437,445), (84,585,591), (84,880,884), (85,132,157),
(85,204,221), (85,720,725),
