Question

In  ΔABC, A = (−4, 1). The internal bisectors the angles B and C are, respectively, x − 1 = 0 and x − y − 1 = 0. Find the coordinates of B and C and the equations of the sides AB and AC.

Answer 1

 See Fig. Let BE and CF be the bisectors of the angles B and C whose equations are, respectively, x − 1 = 0 and x – y − 1 = 0. Suppose M and N are the reflections of the vertex A in the bisectors BE and CF, respectively. Hence, M and N lie on the line BC. Let M = (h, k).

we have

h - 4/1 = k + 1/0 = -2(4 -1)/1  (here k + 1/0 means k = -1)

⇒ h = -2 and k = -1

Hence, M = (-2,-1). Let N = (h', k')Therefore,

h' - 4/1 = k' + 1/-1 = -2(4 + 1 -1/1² + 1²) = -4

⇒ h' = 0 and k' = 3

Hence, N = (0,3) . Therefore, equation of the side BC is

y - 3(3 + 1/0 +2)(x - 0)

2x - y + 3 = 0

Equations of BE is 

x = 1   .....(1)

Equations of CF is

x - y = 1   ...(2)

Equations of BC is 

2x - y = -3   ..(3)

Solving Eqs. (1)–(3), we have B = (1, 5) and C = (−4, −5).