(1) 抛物线与x轴只有一个交点, 则x²－2x + m－1=0的判别式为0：4-4(m-1) = 8-4m = 0, m = 2
(2)取x = 0, y = 1, A(0, 1)
y = (x-1)², B(1, 0)
y = (x-1)² = 1, x = 2, C(2, 1)或 x = 0,此为A点,舍去.
AB² = (1-0)² + (0-1)² = 2
BC² = (1-2)² + (0-1)² = 2
AB = BC
△ABC是等腰三角形
AB的斜率=(1-0)/(0-1) = -1
BC的斜率=(1-0)/(2-1) = 1
二者之积为-1,AB与BC相互垂直,△ABC是等腰直角三角形
(3) y = (x-1)²向下平移4个单位后, 变为y = (x-1)²-4
(x-1)²-4 = 0, x = -1或x = 3(在右半轴,舍去)
E(-1, 0)
x = 0时,y = (x-1)²-4 = -3,F(0, -3)
EF的斜率为(0+3)/(-1-0) = -3
使得△EFP是以EF为直角边的直角三角形, 有两种可能：
(a)E为直角顶点
EP的斜率为(-1)/(-3) = 1/3
EP的方程为: y - 0 = (1/3)(x + 1), y = (x+1)/3
与y = (x-1)²-4联立,得x = -1 (点E,舍去), 或x = 10/3
P(10/3, 13/9)
(b)F为直角顶点
FP的斜率为(-1)/(-3) = 1/3
FP的方程为: y + 3 = x/3, y = x/3 - 3
与y = (x-1)²-4联立,得x = 0 (点F,舍去), 或x = 7/3
P(7/3, -20/9)