See Fig. The slope of BC is
7 - 3/-2 - 1 = -4/3
Therefore, the slope of perpendicular bisector of BC is 3/4 and its equation is
y - 5 = 3/4(x + 1/2)
where M(−1/2, 5) is the midpoint of BC. Since
BC = √(7 - 3)2 + (-2 - 1)2 = 5, we have
Altitude AM = (sin 60°)AB = 5√3/2
The equation of the altitude AM is
y = 3/8(2x + 1) + 5 ...(1)
Now,
75/4 = (AM)2
when x = 2√3 - (1/2) we have
and when x = -2√3 - (1/2) we have
y = 5 - 3√3/2
Hence
Aliter (Using Complex Numbers): B = 1 + 3i and C = −2 + 7i. Rotate CB about C through an angle 60° in anticlockwise sense (see Fig.). Thus
Similarly, if we rotate CB in clockwise sense about C through an angle 60°, we get the second position of A.