Elementary Trigonometry problem

Question

My problem is from Israel Gelfand's Trigonometry textbook.

Page 9. Exercise 7: Two points, $A$ and $B$, are given in the plane. Describe the set of points $X$ such that $AX^2+BX^2=AB^2.$

The attempt at a solution: Since problem asked to describe set of points $X$ such that $AX^2+BX^2=AB^2$, I tried to solve for $X$, and got

$$AX^2+BX^2=AB^2\to X^2(A+B)=AB^2\to X=\sqrt{\frac{AB^2}{A+B}}$$

This got me nowhere though, so I would appreciate some hints on how to approach the problem.

Answer 1

The problem with your attempted solution is the $AX$ does not actually mean $A\times X$, but rather denotes the length of the line segment from $A$ to $X$ (this also applies to $BX$ the length of the segment from $B$ to $X$, and $AB$, the length of the segment from $A$ to $B$).

Now, we can recognize the familiar Pythagorean identity to realize that the triangle formed by $A$, $B$, and $X$ must be a right triangle with hypotenuse the segment from $A$ to $B$. It is commonly known that the locus of such points is the circle with diameter $AB$.

Answer 2

Amazing thought!

In geometry , $AX$ does never imply $$A\cdot X$$

but it implies the distance between the coordinates $A,X$