# How to determine the ratio of AC:AB?

While I was working through a sum related equilibrium of a body I came across this gemetric proof which I was unable to prove. It is given that AG=GB. I need to show that the ratio AC:AB=1:3.

*My working *

I drew a scale diagram to check if the ratio is correct. and then I found that when drawn to scale the ratio fits well. As a start I let the angle AB which makes with horizontal as $$\theta$$ and I let AB as $$2a$$.So far I was only able to determine AX=$$a\cos{\theta}$$ and AD=$$2a\cos{\theta}$$. How should I proceed? Thanks in advance!

P.S: note that red corners are shown to indicated that they are perpendicular. DCX is also a straight line. Please excuse my untidy sketch. I had to draw it on paint.

Draw a perpendicular from $$B$$ to $$AX$$, to say $$F$$. Then $$BF=AD$$. You have similar triangles $$AGX$$ and $$ABF$$, so $$GX=\frac12BF=\frac12AD$$, and $$AG=\frac12AB$$. Then $$\triangle ADC$$ is similar to $$\triangle GXC$$. This yields $$AC=2CG$$. Can you finish?