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So, I was doing high school math to repass my math and there is a problem with which I have quite a bit of difficulty :

Given the triangle A(1,-1), B(3,2) and C(0,3) find the coordinates of M and N (M on AB N on AC) given that MN is parallel to BC and AB=3AM. Verify that BC=3MN.

Sorry if there is any mistakes because I had to translate it. Could anybody either do it or explain the steps (or both) ? Thanks in advance :)

Luca Gervasi
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  • The point $M$ should be on $AB$, or on the line $y=3/2x-5/2$, so consider $M=(a, 3/2a-5/2)$ and use $3AM=AB$ to find $a$, hence the point $M$. – Qurultay Apr 06 '18 at 14:34

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Step 1: draw it.

enter image description here

Step 2: finish it. Since $\frac{BA}{MA}=3$, by the intercept theorem (known in Italy as Thales theorem, which is actually another theorem) $\frac{CA}{NA}=3$, i.e. $MN$ is the image of $BC$ with respect to a dilation centered at $A$ with ratio $\frac{1}{3}$. In particular the coordinates of $M$ are $\left(\frac{5}{3},0\right)$ and the coordinates of $N$ are $\left(\frac{2}{3},\frac{1}{3}\right)$, as the diagram clearly suggests.

Jack D'Aurizio
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