# not very hard and not easy, nice geometry

• Sep 1st 2007, 07:28 AM
Ununuquantium
not very hard and not easy, nice geometry
An angle is given (<180 degree) with a vertex X and a point P placed inside this angle. Points A and B lie on different arms of this angle, and XA=XB and the sum PA+PB is the smallest. I ask: when this sum is the smallest? Maybe the angles APX and BPX are equal??

It is not so easy, that's why i placed it here. Can anybody help me?
• Sep 1st 2007, 08:26 AM
red_dog
I assume that $\displaystyle A,B$ are fixed points such that $\displaystyle XA=XB$ and $\displaystyle P$ is a variable point such that $\displaystyle PA+PB$ is smallest.
Then $\displaystyle P\in[AB]$ and in this case $\displaystyle PA+PB=AB$
If $\displaystyle P\not\in[AB]$, then $\displaystyle PA+PB>AB$.
$\displaystyle P$ can be anywhere on the segment $\displaystyle AB$, so is not necessary that $\displaystyle \widehat{APX}=\widehat{BPX}$
Also, the condition $\displaystyle XA=XB$ is not necessary.
• Sep 1st 2007, 08:46 AM
Ununuquantium
Quote:

Originally Posted by red_dog
I assume that $\displaystyle A,B$ are fixed points such that $\displaystyle XA=XB$ and $\displaystyle P$ is a variable point such that $\displaystyle PA+PB$ is smallest.

No, you cannot do that. Firstly we place poin P inside the angle and than we must construct such points A and B. The question is formed properly
• Sep 1st 2007, 11:35 AM
ticbol
Quote:

Originally Posted by Ununuquantium
An angle is given (<180 degree) with a vertex X and a point P placed inside this angle. Points A and B lie on different arms of this angle, and XA=XB and the sum PA+PB is the smallest. I ask: when this sum is the smallest? Maybe the angles APX and BPX are equal??

It is not so easy, that's why i placed it here. Can anybody help me?

If (PA + PB) is the smallest, then (PA + PB) must be a straight line. The shortest distance between two points, A and B here, is a straight line.

That means P is anywhere along the line AB.

That means then that angles APX and BPX are supplementary----their sum is 180 degrees.

Therefore, (PA + PB) is smallest when angles APX and BPX are supplementary. ---------answer.