Question

Ellen has property whose boundary lines form a triangle, as shown in the diagram. Her house lies at Vertex A and she is going to put a bunkhouse for the ranch hands somewhere along segment BC. She wants the placement to be such that CA/CD=B/A. What must be true about segment AD so that the proportion CA/CD=BA/BD holds

Answer 1

Given that in a ΔABC, Ellen house lies at vertex A and bunkhouse is put somewhere along BC, let it be D.

Please refer the diagram attached.

We are also given that D must be placed such that $\frac{CA}{CD}= \frac{BA}{BD}$ holds true.

Let us re write the above equation such that CD and BD lie on one side.

That is $\frac{CA}{BA}= \frac{CD}{BD}$

Which is nothing but angle bisector theorem.

Hence AD must bisect the angle A then only the given condition will hold true.