Given a point P in the interior of an angle with vertex O, find points A and B on the sides of the angle so that OA=OB and the sum PA+PB is minimal
We want OA = OB, so
So the coordinates of A and B are:
Let the coordinates of point P be simply .
Now you want to set up an expression for the the distance
Now you take it from here.
Than you very much Dan.
Could you please prove that angles BPO and APO are equal?
I cannot do it using coordinate system (I like classical geometry solutions , and I am not so good in coordinate system - I just do not like counting) So I thaught about normal solution, maybe using symmetry ? I think this problem is not so hard to do it using coordinate system - but if You want to use it Dan, please show me whole solution with proving that angles BPO and APO are equal. Could you please help me with whole solution ??
Here's a hint for what to do with your distance equation in case it comes in handy:
Solve to get the critical points.
Here's a more geometric approach to the problem, which answers the question about angles.
Rotate the diagram through an angle about the point O (where is the angle AOB), so that A goes to A', B goes to B' and P goes to P'. Superimpose the rotated diagram on the original diagram. Then A' coincides with B (because OA=OB). We want to minimise AP+PB = BP'+PB. But this is easy: the minimum obviously occurs when PBP' is a straight line.
So the construction goes as follows: rotate P through the angle AOB about the point O, to get P'. Then B is the point where PP' crosses the line OB.
The triangle OPP' is isosceles (because OP=OP'). Therefore the angles OP'B and BPO are equal. But OP'B=OPA. So the angles BPO and OPA are equal.
. . . . . . . . .
Ok, I understand - but tell me - You take hypothetical pionts A and B, and then you rotare everything and you say that this sum AP+PB is the smalles when PBP' is a straight line?
I mean the construction is like that:
We have an angle with vertex O and point P inside. We put hypothetical pionts A and B such that OA=OB and THEN we rotate "everything" - and then we say that this sum AP+PB is the smalles when PBP' is a straight line?
And thank You very much