# Triangle Construction Problems

## Question

Given triangle ABC, pick any point P that lies in its interior. Through P, construct the three lines parallel to the sides of the triangle, as shown.
The lengths of the segments in bold are labeled.
Prove: (x/BC)+(y/AC)+(z/AB)=1

[Hint: The problem with these proofs is that its not exactly clear where to start. There are lots of similar triangles in the diagram, but which ones do we need? Well, look at the equation you need to prove; the segment lengths involved here are from the three middle triangles and the big triangle. Are any of these triangles similar? Why? Also, in whatever problem you happen to be working on from now on, you should be looking for our old friends lurking about; for example, see the parallelograms? and as usual, write equations, use algebra]
No angle chasing is necessary. Fiddle with angles to get similar triangles.

## Solution Preview

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There are many parallel lines in this triangle that should allow us to use properties of similar triangles. This is a tricky problem and it is easy to get confused.

We need to show that: x/BC + y/AC+z/AB=1.

Let us take a look at the quantity 1- z/AB. We need to show that it is equal to x/BC + y/AC.
Now 1-z/AB=(AB-z)/AB.
And AB-z is the length of two line segment, one from A to the line segment z, that I will call AZ
and the other ZB.
So (AB-z)/AB = AZ/AB + ZB/AB....
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