intruiging geometry problem

okay so here's the problem that i found on the net...

"in the picture, AD is angle bisector if AB + AD = CD and AD + AC = BC ,

angle ACB = 20 degree,"

find angle ABC

here's the link to the picture (http://img694.imageshack.us/img694/6562/pb3.png)

and there are hints onto solving this problem as well... but i still can't doooo it.

"AD is a bisector implies

AB : AC = area(ABD) : area(ADC) = BD : DC

Use this to express BC/AB and hence cos(angle ACB) in

terms of k = AC/AB. Now, what is sin(angle ABC) in terms

of k and sin(angle ACB)?"

TIA :)

Re: intriguing geometry problem

Quote:

Originally Posted by

**avitos** okay so here's the problem that i found on the net...

"in the picture, AD is angle bisector if AB + AD = CD and AD + AC = BC ,

angle ACB = 20 degree,"

find angle ABC

here's the link to the picture (

http://img694.imageshack.us/img694/6562/pb3.png)

and there are hints onto solving this problem as well... but i still can't doooo it.

"AD is a bisector implies

AB : AC = area(ABD) : area(ADC) = BD : DC

Use this to express BC/AB and hence cos(angle ACB) in

terms of k = AC/AB. Now, what is sin(angle ABC) in terms

of k and sin(angle ACB)?"

Following the hint, let , so that and . Then

.

But also

Comparing those two equations, you see that and hence

Thus and Now the cosine formula in triangle ABC tells you that

Divide top and bottom by to get

(after a bit of simplifying). Therefore

Finally, the sine rule tells you that , from which

and so angle (Whew)