The context is construction of three circles with different radii so that they touch each other externally using a graduated ruler and a compass.
I have done it by drawing a triangle where each side is the sum of the radii of distinct combination of two circles and then drawing the circles by taking the vertex as the centre of a circle.
My question is why does my method work and is this the standard method to do the construction in context because i could not find anything about it on the internet.
Your method works well. If you draw the line segments between each pair of centers they are composed of two radii, which justifies your construction.