Locus Theorem 5.

I am back. I apologise to keep you waiting. It is obvious at the intuitive level that orange line is equidistant from x axis and line x=2. The first proof I got in my mind was to make axes coordinate transformation, so x=2 will coincide with y axis, this will simplify the proof.

Graph from previous post

But, wait a minute… We should understand that this problem was posed to a 9 year old child and it was part of a test. So you should tackle this problem within a couple of minutes. So you either should have a very deep understanding and intuition, which is quite unlikely at this age… Or you should know a quick “recipe” for this sort of problems. Locus Theorem 5 is what we need to formulate our “recipe”.

In geometry, a locus (plural: loci) is a set of points whose location satisfies or is determined by one or more specified conditions. As my daughter just started to learn Latin, she might know that in Latin, the word locus means place.

LOCUS THEOREM 5

The locus of points equidistant from two intersecting lines, L and Y, is a pair of bisectors that bisect the angles formed by L and Y .

Illustration for locus theorem

From the illustration above we can notice two intersecting lines, L and Y, and two lines which are equidistant from them have common intersection point.

MATHEMATICAL “RECIPE”

Wow… I am formulating first mathematical recipe in this blog (btw this is my first first experience :)).

If you have several equations of the lines and you need to choose the one which is equidistant to some other lines, you need to check whether they have common intersection point or not.

The END :)