Question

Hello,

Can someone show me the steps to solve this college geometry practice question?

Be be legible, thank you!

Probliem 1.a. The ezternal ondl tbachors of a tane d r oppete sides at three collinear points. x'

Answer 1

ABC is a triangle, AX^', BY^' and CZ^' are angle bisectors(external) of A, B, and C respectively. From X^' draw the line parallel to AB intersecting the extension of AC at D. The two triangle BAX^' and DAX6' are isosceles and congruent, hence AB/AC = DX^'/DC = DA/DC = X^' B/X^' C Thus the foot of an external bisector cuts the opposite side at a ratio equal to the ratio of the two adjacent sides. Therefore if BX^' and CZ^' are the remaining two external bisectors of ABC, we have X^' B/X^' C Y^' C/Y^'A Z^' A/Z^' B = c/b a/d b/a = 1 Therefore by Menelau's theorem, the field of the external bisectors are collinear. (where AB = c AC = b BC = a)