# A triangle has vertices at #A(a,b )#, #C(c,d)#, and #O(0,0)#. What are the endpoints and length of the perpendicular bisector of AC ?

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Hey, this got featured before it was really done. It's still not done.

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This is another in my series where I generalize one of these old questions.

"The triangle's perpendicular bisectors" is not a term we hear that often. Each side has a perpendicular bisector, the perpendicular through the midpoint. It will occasionally intersect the opposite vertex (in which case it's the altitude of an isosceles triangle) but usually it will intersect exactly one of the other sides.

I've rewritten the question to put one vertex at the origin and to just ask for the perpendicular bisector of the opposite side.

The sides of the triangle are given parametrically as

The perpendicular family to **bisector** is through the midpoint of AC:

That meets

Two equations in two unknowns.

We're more interested in

The roles aren't quite symmetrical so we repeat the process for the meet of the bisector and

Yes I know this is getting long Socratic. It's an involved problem.

The numerators are the same in

~~~~~~~

We want to solve

Let's first assume

We're assuming

What about when

How about

For

Unless

Still not there.

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We need to show the denominators have opposite signs.

~ ~~~~~~

The triangle inequality says

The dot product and the sum of squares have a close relationship. We expand

We still haven't shown we pass through at most one side or the vertex. It remains to show

Still more to do.

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If

If

If

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My other answer got featured before I finished it. I'm looking for a simpler way that I can actually finish.

Let's start with a simpler problem. Triangle OPI, coordinates

Solution: The foot of the bisector is the midpoint of

If

If

If

That seems simple enough. We want to map

The general rotation, scaling, translation is

We mapped O to P. Unknowns

Let

The remaining difficult issue is the inverse mapping so we can get the other endpoints

The almost inverse is

Transforming our original solution,

If

If

If

Describe your changes (optional) 200