MHB Find Point in R2: $a(-5,2),b(12,9)$

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To find a point \( c \) in \( \mathbb{R^2} \) such that \( ac = bc \) and \( ac \perp bc \) for points \( a(-5,2) \) and \( b(12,9) \), the solution involves determining the perpendicular bisector of segment \( ab \), which contains all points equidistant from \( a \) and \( b \). The conditions imply that \( c \) must also form a square with points \( a \) and \( b \), centered at the midpoint \( (3.5, 5.5) \). The two possible coordinates for \( c \) are derived as \( (0,14) \) and \( (7,-3) \). This geometric approach offers a more efficient solution than the initial algebraic method.
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Let $a(-5,2),b(12,9)$ be two points in $\mathbb{R^2}$ find a point $c\in \mathbb{R^2}$ such that

$ac = bc$ and
$ac \perp bc$

I have a long solution I made two equations ( I considered $c (x,y)$ )

$$ ac = bc $$
$$ \sqrt{ (x +5)^2 + (y-12)^2 } = \sqrt{(x -12)^2 + ( y - 9)^2 }$$

and

$$ \frac{y-9}{x-12} \cdot \frac{y-2}{x+5} = -1 $$

and solve them by substitution

is there any faster solution shorter ?
Thanks in advance.
 
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Amer said:
Let $a(-5,2),b(12,9)$ be two points in $\mathbb{R^2}$ find a point $c\in \mathbb{R^2}$ such that

$ac = bc$ and
$ac \perp bc$

I have a long solution I made two equations ( I considered $c (x,y)$ )

$$ ac = bc $$
$$ \sqrt{ (x +5)^2 + (y-12)^2 } = \sqrt{(x -12)^2 + ( y - 9)^2 }$$

and

$$ \frac{y-9}{x-12} \cdot \frac{y-2}{x+5} = -1 $$

and solve them by substitution

is there any faster solution shorter ?
Thanks in advance.
That solution looks fine (and not really too long). An alternative method would be to draw a diagram and do some geometry.

The set of points equidistant from $a$ and $b$ is the perpendicular bisector of $ab$. So $c$ must lie on that line. In fact, there are two possible positions for $c$, I'll call them $c_1$ and $c_2$.

The other condition, $ac \perp bc$, implies that the points $a,c_1,b,c_2$ are the vertices of a square centred at the midpoint of $ab$, $(3.5,5.5)$. The vector from there to $b$ is $(12,9) - (3.5,5.5) = (8.5,3.5)$. A perpendicular vector of the same length is $\pm(-3.5,8.5)$. So the possible positions for $c$ are $(3.5,5.5) \pm (-3.5,8.5)$. That gives $c_1 = (0,14)$ and $c_2 = $(7,-3)$.
 

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Great solution thanks :)
 
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