MHB Show Equivalence of AB Parallel to XY | Wondering

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The discussion revolves around proving that line segment AB is parallel to line segment XY if and only if certain ratios involving points X, Y, and Z in triangle ABC hold true. Participants explore the relationships between the triangles formed by these points, noting that if AB is parallel to XY, triangles AZB and ZXY are similar, leading to proportional side lengths. There is debate about whether the original statement should involve equality or a multiplicative relationship between the ratios. The conversation also touches on using vector methods to establish the necessary conditions for parallelism. Ultimately, the participants are working towards a clearer understanding of the geometric relationships and conditions for parallel lines.
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Hey! :o

We have the triangle ABC. We have that $X\in AC$, $Y\in BC$ and $Z=AY\cap BX$, where $X,Y\neq A,B,C$. I want to show that AB is parallel to XY iff $\frac{|CX|}{|CA|}=\frac{|ZB|}{|ZX|}=1$.

Could you give me a hint what we have to show?

When AB is parallel to XY then we have that then AZB and ZXY are congruent. What do we get from that? (Wondering)
 
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mathmari said:
Hey! :o

We have the triangle ABC. We have that $X\in AC$, $Y\in BC$ and $Z=AY\cap BX$, where $X,Y\neq A,B,C$. I want to show that AB is parallel to XY iff $\frac{|CX|}{|CA|}=\frac{|ZB|}{|ZX|}=1$.

Could you give me a hint what we have to show?

Hey mathmari! (Smile)

If we pick BX and AY to be medians, we have $\frac{|CX|}{|CA|}=\frac 12 \ne \frac 21 =\frac{|ZB|}{|ZX|} \ne 1$.
Can it be that $\frac{|CX|}{|CA|} \cdot \frac{|ZB|}{|XZ|}=1$ was intended?

mathmari said:
When AB is parallel to XY then we have that then AZB and ZXY are congruent. What do we get from that?

Aren't $AZB$ and $ZXY$ similar instead of congruent?

Anyway, yes, I believe we should use that if $AB\parallel XY$, that then $AZB$ and $ZXY$ are similar.
That means that the corresponding sides have the same ratio.
Btw, $ABC$ and $XYC$ are also similar. (Thinking)
 
I like Serena said:
If we pick BX and AY to be medians, we have $\frac{|CX|}{|CA|}=\frac 12 \ne \frac 21 =\frac{|ZB|}{|ZX|} \ne 1$.
Can it be that $\frac{|CX|}{|CA|} \cdot \frac{|ZB|}{|XZ|}=1$ was intended?
I don't know. Maybe. The exercise statement is as in #1.
I like Serena said:
Aren't $AZB$ and $ZXY$ similar instead of congruent?

Anyway, yes, I believe we should use that if $AB\parallel XY$, that then $AZB$ and $ZXY$ are similar.
That means that the corresponding sides have the same ratio.
Btw, $ABC$ and $XYC$ are also similar. (Thinking)

From the fact that the triangles $AZB$ and $ZXY$ are similar, we get that $\frac{BZ}{ZX}=\frac{AZ}{ZY}=\frac{AB}{XY}=m_1$.

From the fact that the triangles $ABC$ and $XYC$ are congruent, we get that $\frac{XY}{AB}=\frac{YC}{BC}=\frac{XC}{AC}=m_2$.

Since $m_1=\frac{AB}{XY}$ and $m_a=\frac{XY}{AB} \Rightarrow \frac{AB}{XY}=\frac{1}{m_2}$ we get that $\frac{BZ}{ZX}=\frac{AC}{XC}$, right? (Wondering)

Maybe we have to show the relation without equal to $1$ ? (Wondering)
What about the other direction? (Wondering)

Does the other direction would hold if we would have $\frac{BZ}{ZX}=\frac{AC}{XC}$ without equal to $1$ ? (Wondering)
 
mathmari said:
Does the other direction would hold if we would have $\frac{BZ}{ZX}=\frac{AC}{XC}$ without equal to $1$ ? (Wondering)

Suppose it holds that $\frac{BZ}{ZX}=\frac{AC}{XC}=:m$.

We consider the triangles $AZB$ and $XZY$. We have that $BZ=m\cdot ZX$ and the angle of $Z$ in both triangles are the same.
We consider the triangles $ABC$ and $XCY$. We have that $AC=m\cdot XC$ and the angle of $C$ in both triangles are the same.

Can we get something from here? (Wondering)
 
mathmari said:
I don't know. Maybe. The exercise statement is as in #1.

From the fact that the triangles $AZB$ and $ZXY$ are similar, we get that $\frac{BZ}{ZX}=\frac{AZ}{ZY}=\frac{AB}{XY}=m_1$.

From the fact that the triangles $ABC$ and $XYC$ are congruent, we get that $\frac{XY}{AB}=\frac{YC}{BC}=\frac{XC}{AC}=m_2$.

Since $m_1=\frac{AB}{XY}$ and $m_a=\frac{XY}{AB} \Rightarrow \frac{AB}{XY}=\frac{1}{m_2}$ we get that $\frac{BZ}{ZX}=\frac{AC}{XC}$, right?

Yes. (Nod)

mathmari said:
Maybe we have to show the relation without equal to $1$ ?

If so, then we also need to invert one of the fractions.
Either way, the result would be the same as replacing the $=$ sign by a multiplication.
mathmari said:
What about the other direction?

Does the other direction would hold if we would have $\frac{BZ}{ZX}=\frac{AC}{XC}$ without equal to $1$ ?

mathmari said:
Suppose it holds that $\frac{BZ}{ZX}=\frac{AC}{XC}=:m$.

We consider the triangles $AZB$ and $XZY$. We have that $BZ=m\cdot ZX$ and the angle of $Z$ in both triangles are the same.
We consider the triangles $ABC$ and $XCY$. We have that $AC=m\cdot XC$ and the angle of $C$ in both triangles are the same.

Can we get something from here?

I don't see how.

As I see it, either we have to assume that $XY \nparallel AB$ and deduce that the fractions are different.
Or we assume the fractions are the same and deduce that $XY \parallel AB$.
I haven't figured out how to do that yet though.
Perhaps we could do it with vectors instead of geometry properties.
 
For the other direction we have that $\frac{BZ}{ZX}=\frac{AC}{XC}$.

Let $Y'\in BC$ so that $AB\parallel XY'$.
Then $Z'$ is the intersection point of $AY'$ and $BX$.
We have as in the other direction that the triangles $AZB$ and $Z'XY'$ are similar and so it holds that $\frac{BZ'}{Z'X}=\frac{AZ'}{Z'Y'}=\frac{AB}{XY'}$.
The triangles $ABC$ and $XY'C$ are also similar and so it holds that $\frac{XY'}{AB}=\frac{Y'C}{BC}=\frac{XC}{AC}$.
So, we get that $\frac{BZ'}{Z'X}=\frac{AC}{XC}$.
Since $\frac{BZ}{ZX}=\frac{AC}{XC}$, we get that $\frac{BZ'}{Z'X}=\frac{BZ}{ZX}$. Do we conclude from here that $Z'=Z$ and so that $Y'=Y$ ? (Wondering)
 
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