Find A Point on Line with Minimal (abstract?) Information Reference Page
I am creating a program that can draw sewing patterns based on a person's measurements. The reference material I have assumes you are working on paper, thus some of the problems I am having. I am writing a function that creates points that I can then use to draw the pattern. In some cases the points are simply reference to draw other points and won't actually be displayed. Also, just to make things more complicated, I am switching from inches (reference material) to centimeters.
The link is to the reference page I'm working with to better understand what I'm talking about (site wouldn't let me upload it). I also put in all the points that I've calculated so far.
I wasn't having any problems until I started working with the angled lines. My first roadblock was Figure 2 line B-G. We used the pythagorean theorem with line A-C length (17.78), line B-G length (43.498) to come up with the line length to get the y coordinate for G. If that was a mistake, please let me know. I was going to use that same process for the I point.
Assuming that the G coordinate is correct, now I have to find the H coordinate. I have absolutely no idea how to do this. The only measurement I have is that it's 22.86cm from G to H. Since I have the full triangle from calculating G, I can now probably get the slope of B-G, if necessary??
I would prefer to understand how to do this, as opposed to just a solution. I am likely to encounter similar situations as I proceed with this project. 
Any assistance is greatly appreciated.
 A: You calculated the $y$ coordinate of $G$ correctly (it is $43.18-\sqrt{43.498^2-17.78^2}=3.482$).
$BG$ is the hypotenuse of the right triangle with two vertices $G$, $B$, and the third right-angled vertex, which I'll call $Z$, on the line segment  $EB$. 
You know all the side  lengths here. Namely 
$\ \ \ \ \ |GZ|=43.18-3.482=39.7$ 
$\ \ \ \ \ |GB|=43.498$
and (from figure 1.)
$\ \ \ \ \ |ZB|=|AC|=17.78$.   
To find the coordinates of $H$, we will use the properties of similar triangles: 
let $X$ be the point on the line segment $EB$  so that $HX$ is perpendicular to $EB$.  The triangles $\triangle GZB$  and $\triangle HXB$ are similar.  So you know corresponding ratios of sides lengths are equal. For instance:
$$
{|GB|\over |ZB|}={|HB|\over |XB|}.
$$
Substitute the known values  $|GB| $, $|ZB| $, and $|HB|= |GB|-|GH|$  into the above equation and then solve for  the value of $|XB|$ (which is the horizontal coordinate of $H$).
A similar computation allows you to find the vertical coordinate of $H$ (or you can use the Pythagorean Theorem applied to $\triangle HXB$, here, since at this point you know two side lengths). 
