How can you prove that the internal bisectors of a triangle are concurrent, using vector methods only?
Thank you guys!
First you need a way of describing the points on the internal bisector of an angle. Suppose you start at point x with the angle having direction vectors u and v. The internal bisector consists of all points z for which x and z are opposite vertices of a rhombus with edges along u and v: that is, with edges tu and tv for some scalar t. So z = x + tu + tv.
One internal bisector of a triangle with vertices at a,b,c is at a with edges having direction vectors (b-a),(c-a). So that bisector consist of points a + r(b-a) + r(c-a) for scalar r. The other two are b + s(a-b) + s(c-b) and c + t(a-c) + r(b-c).
You need to show that there is a point which lies on all three lines: that is, that there are values r,s,t for which these three points are the same, that is,
a + r(b+c-2a) = b + s(a+c-2b) = c + t(a+b-2c).
Can you find such values of r,s,t? (Hint: would you expect the values of r,s,t to be the same or different?)