Question

D, E, F are points dividing side vector(BC, CA, AB) of a triangle ABC in the ratio 2:3, 1:2 and 3:1 respectively. Show that the lines vector(AD, BE, CF)  are concurrent and hence find the position vector of their point of intersection.

Answer 1

By using section formula we can obtain required result

If vector(d, e, f) are position vector of points D, E & F respectively then, by section formula

Equation of line AD is vector r = vector(a + t(d - a)

Equation of line BE is vector r = vector b + m(e - b)

For intersection of vector AD and vector BE  we need that

∴ 1 - t = m/3; 3t/5 = 1 - m; 2t/5 = 2m/3

∴ t = 5/6, m = 1/2

The existence of t and m assures the intersection of  vector AD and vector BE.

The point of intersection is