Demonstrate equality: ON = 2m/m-3 in math exercise

Question

I'm actually getting stuck with a part of a quite tricky math exercise using Thales theorem (I've got difficulties with Thales theorem).

In this exercise,you have a right handed Cartesian coordinate system (O; i, j).A is a point of(AM) line which have the coordinates of (3;2). M is a point of X axis of coordinate (m ; 0) with m > 3. The line (AM) intersects Y axis in N.

You have to demonstrate that ON = 2m/m-3

Thanks for your answers

PS: Sorry if I made mistakes with my english, I'm french and I'm only high school level in english.

Answer 1

Let the orthogonal projection of A onto x-axis by T, which has coordinate (3,0). Then we can conclude $\frac{AT}{NO}=\frac{MT}{MO}$. Then plug in the length and you can get the result.