Right Angles

[JohnBoher]

Given:
Triangle ABC,
Lines AE, AF trisect side BC, with AE to the left of AF, and points E, F on line BC,
Points G, H bisect sides AC, AB respectively.

Are the angles formed by lines AF and GH right angles?

I am stuck on this problem, and so far I proved triangle AHG similar to ABC by a ratio of 1:2, since G, H bisect AB and AC (side splitter theorem).

 

[pka]

Given: Triangle ABC,
Lines AE, AF trisect side BC, with AE to the left of AF, and points E, F on line BC,
Points G, H bisect sides AC, AB respectively.
Are the angles formed by lines AF and GH right angles?.

Unless you can post a diagram, I doubt that you will get any answers.

What you have posted makes little sense to me for one.
As I read it \(\displaystyle \overline{GH}~\&~\overline{AF}\) form a vertical pair of angles.
It is hardly likely they are right angles.

 

[JohnBoher]

I apologize for the lack of clarity, here is the diagram as it is on my paper:



It certainly seems like this angle would be 90 degrees, I just can't figure out why this is so. Can I just assume that it is 90 for the rest of the problem, as that is one possible value? What I needed to find out was the ratio of the areas of triangles AHG and AFE, which turns out to be 3 : 4 if I assume its a right angle (which is the correct answer to the problem).

 

[pka]

I apologize for the lack of clarity, here is the diagram as it is on my paper:

View attachment 4087

It certainly seems like this angle would be 90 degrees, I just can't figure out why this is so. Can I just assume that it is 90 for the rest of the problem, as that is one possible value? What I needed to find out was the ratio of the areas of triangles AHG and AFE, which turns out to be 3 : 4 if I assume its a right angle (which is the correct answer to the problem).

Note that \(\displaystyle \overline{GH}\|\overline{CB}\) and is one-half its length.
That means there are many many similar triangles in that diagram.

As for area, \(\displaystyle \Delta ACB\) is eight times \(\displaystyle \Delta AGH\).

 

[wjm11]

As for area, [FONT=MathJax_Main]Δ[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]C[/FONT][FONT=MathJax_Math]B[/FONT] is eight times [FONT=MathJax_Main]Δ[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]G[/FONT][FONT=MathJax_Math]H[/FONT].

I think you meant "four" times... ?