Question 1200535
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This problem has an underwater stone,  which is usually unseen to many people.


This underwater stone is that the problem is OVER-defined.


Indeed,  the condition giving coordinates of  TWO  tangent points is  EXCESSIVE: 
one tangent point is just enough and it defines the second tangent point
by an  UNIQUE  way.


In couple of words,  I will explain  WHY  the problem is over-defined.


<pre>
    Indeed, we know that the center must lie on the bisector of the angle,
    concluded by the given lines.

    From the other side, the center of the circle must lie on the perpendicular
    to one of the given lines at the tangency point - so the center
    of the circle is the intersection of the angle bisector and the 
    perpendicular to one of the tangency line at the tangency point.
</pre>


What are the consequences from the fact that the problem is over-defined ?


The consequence is that when the center is found as the intersection point
of two perpendiculars to the given lines at the tangency points,
the person, who solves the problem, &nbsp;MUST &nbsp;check that the distance
from the intersection point to the given tangency points &nbsp;IS &nbsp;THE &nbsp;SAME:



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It will guarantee that the condition of the problem 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;is self-consistent and is not self-contradictory.



Without such a check, &nbsp;the solution is formally incomplete; 
it is completed &nbsp;ONLY &nbsp;when the check is done.



Fortunately, &nbsp;in our case &nbsp;(it is easy to check) &nbsp;the distance from the intersection 
point &nbsp;(1,0) &nbsp;to the given tangency points is the same: &nbsp;it is equal to  &nbsp;{{{sqrt(5)}}}.



--------------------



<U>Comment from student</U> : &nbsp;&nbsp;It's easier of the author provided a graph.



<U>My response</U> : &nbsp;&nbsp;In &nbsp;Geometry, &nbsp;the plots are never considered as a proof 
or a tool to make a proof: &nbsp;the plots work and are used for visualization, &nbsp;ONLY.


Especially, &nbsp;in this problem, &nbsp;where the radius is an irrational number &nbsp;{{{sqrt(5)}}}, 
and you can not distinct visually &nbsp;{{{sqrt(5)}}} &nbsp;from {{{sqrt(5.1)}}}.


So, &nbsp;your attempt to object or to argue my conception is invalid.