complicated triangle

[logistic_guy]

here is the question

In the diagram, \(\displaystyle N\) is the incenter of \(\displaystyle \Delta ABC\). Find \(\displaystyle NF\).




my attemb
how to know if \(\displaystyle EB\) is a straight line or not
it appear to me not straight i can't start my analize if i'm not sure of that

 

[Dr.Peterson]

here is the question

In the diagram, \(\displaystyle N\) is the incenter of \(\displaystyle \Delta ABC\). Find \(\displaystyle NF\).

View attachment 38964


my attemb
how to know if \(\displaystyle EB\) is a straight line or not
it appear to me not straight i can't start my analize if i'm not sure of that

E, N, and B are clearly not collinear, and can't be, since [imath]EN\perp AC[/imath] and [imath]\angle ABN\cong\angle CBN[/imath]. (But if you didn't see that, then the answer would be that you can only use facts that you know. You can't just guess. And if they wanted you to know something that is not clear from the picture, they would be required to tell you in words.)

What do you know about the meaning of "incenter"? (If you know nothing, look it up!)

If that's not enough, compare triangles NFB and NGB.

 

[logistic_guy]

E, N, and B are clearly not collinear, and can't be, since [imath]EN\perp AC[/imath] and [imath]\angle ABN\cong\angle CBN[/imath]. (But if you didn't see that, then the answer would be that you can only use facts that you know. You can't just guess. And if they wanted you to know something that is not clear from the picture, they would be required to tell you in words.)

great information. this already give the idea of the incenter before i read about it

What do you know about the meaning of "incenter"? (If you know nothing, look it up!)

i don't. i read about it
In a triangle, the incenter is the point where the angle bisectors of the triangle intersect. The angle bisectors are the lines that divide each of the interior angles of the triangle into two equal parts.

i think this information tell me i've \(\displaystyle 6\) right triangles inside the big triangle each pair share the same angle is equal
this mean\(\displaystyle \Delta NFB\cong\Delta NGB\)
which mean \(\displaystyle NF = NG\)
\(\displaystyle NG^2 + 12^2 = 13^2\)
\(\displaystyle NG^2 + 144 = 169\)
\(\displaystyle NG^2 = 169 - 144 = 25\)
\(\displaystyle NG = \sqrt{25} = 5\)
then \(\displaystyle NF = NG = 5\)

is my analize correct?