The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D.
Find the coordinates of D and the length AD.
We know that angle bisector divides opposite side in ratio of other two sides
⇒ D divides BC in ratio of AB : AC A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2)
AB=√42+52+32
=√16+25+9=√50+5√2
AC = √12+12+42
=√1+1+16=√18=3√2
AD is the internal bisector of ∠BAC
∴BDDC=ABAC=53
Thus, D divides BC internally in the ratio.
∴D=(5×4+3×15+3,5×3+3(−1)5+3,5×2+3×35+3)
⇒D=(238,128,198)
∴AD=√(5−238)2+(4−128)2+(6−198)2
=√172+202+29282
=√289+400+84182=√15308