What geometry theorem is used in this figure?

[FactChecker]

You are wrong. The drawing is correct. You should NEVER assume things. If the drawing does not say it is a right triangle, then do not assume it is.

Ok.
Should we assume that the baseline is one straight line or is there an angle at the vertex between the 4 and 6 length lines?
Should we assume that all those lines to the top meet at the same point? It looks a little like the line farthest to the right comes in at a different point.
Of course, I am being sarcastic. The point is that the easier you can make it for us, the more likely it is that you will get good help.

 

[phinds]

@barryj you should learn the first rule of conversation/debate: when you find yourself in a hole, stop digging.

 

[barryj]

Ok.
Should we assume that the baseline is one straight line or is there an angle at the vertex between the 4 and 6 length lines?
Should we assume that all those lines to the top meet at the same point? It looks a little like the line farthest to the right comes in at a different point.

No, there is a 120 degree angle at the 4/6 intersection and the lines do not meet at the same point. They are separated by 2 inches. This thread is getting silly. I am gone!

 

[FactChecker]

No, there is a 120 degree angle at the 4/6 intersection and the lines do not meet at the same point. They are separated by 2 inches. This thread is getting silly. I am gone!

Of course, I was being sarcastic, but I actually thought it was a right angle until I did the calculation and didn't get 12.

 

[barryj]

Of course, I was being sarcastic, but I actually thought it was a right angle until I did the calculation and didn't get 12.

You had best get another calculator.

 

[FactChecker]

You had best get another calculator.

My calculator is fine. If that was a right angle, the length of $x$ would be 12.8062484748657.
(And the angle would not be bisected to give a partitioning of 4 and 6. Although, I didn't calculate what they should be.)

 

[berkeman]

Thread closed temporarily for Moderation...

 

[berkeman]

After some thread cleanup, the thread will remain closed. Thank you everybody for helping the OP with his question.

 

[barryj]

[Mentor Note -- OP has requested that this clarification post be added to the end of this closed thread]

The moderator closed part 1 before I could post the requested theorem. I have attached the figure and a copy of the theorem from a geometry book for those that are interested.