Constant quotient

Constant product

$\textsf{\LARGE B}$

Constant difference

Constant sum

Checking

$\textsf{\LARGE A}$

POINT - POINT

When the sum of the distances of the points of the searched locus 
to points A and B is constant, we obtain an ellipse.

When the difference of the distances of the points of the searched locus 
to the points A and B is constant, we obtain a branch of a hyperbola. 
(In the case that the constant is 0, we obtain the perpendicular bisector.)

When the product of the distances of the points of the searched locus 
to points A and B is constant, we obtain a Cassini oval. 
If the constant coincides with the square of half the distance AB, 
we obtain a Bernoulli lemniscate.

When the quotient of the distances of the points of the searched locus
 to points A and B is constant, we obtain, surprisingly, a circle. 
(In the case that the constant is 1, we obtain the perpendicular bisector.)

Activate offset

op4 = false
op3 = false
r = 3
B: (4, -1)
I: (1, 0)
s = Line(B, I)
op2 = false
op1 = true
C: (1.4, 3.8)
h = 5 (slider from 0 to 5, step 0.01)
k = 5 (slider from -8 to 8, step 0.01)
comprobar = false
A: (0, 0)
s1 = Segment(C, A)
s2 = Segment(B, C)
texto5 = If(op3, "DEGREE 4", "DEGREE 2")
kk = If(op1, Distance[C, B] + Distance[C, A], If[op2, Distance[C, A] - Distance[C, B], If[op3, (Distance[C, B] * Distance[C, A]), Distance[C, A] / Distance[C, B]]])
textoComprueba: "k = " + kk + ""
ci1:=XA + XB = k
ci2':=((4 * XB^(2)) * XA^(2)) = (k^(2) - XA^(2) - XB^(2))^(2)
ci2:=XA - XB = k
ci3:=(XA * XB) = k
ci3':=(XA^(2) * XB^(2)) = k^(2)
ci4:=XA = (k * XB)
ci4':=XA^(2) = (k^(2) * XB^(2))
op5 = false
h': 5 - h
sA = Circle(A, If[op1, k - h', If[op2, k + h', If[op3, k / h', (k * h')]]])
sB = Circle(B, h')
texto6: "XA " + (If[op1, "+", op2, "−", op3, "·", "/"]) + " XB = k"

Tool: OpSuma
Elige O y luego dos puntos
Input: O, A, B
Output: O'

C = Midpoint(A, B)
O' = Mirror(O, C)

Tool: OpResta
Elige O, el primer punto y el segundo punto
Input: O, A, B
Output: O'

B' = Mirror(B, O)
C = Midpoint(A, B')
O' = Mirror(O, C)

Tool: OpProducto
Elige O e I, y luego dos puntos
Input: O, I, A, B
Output: E

I' = Mirror(I, O)
B' = Mirror(B, O)
c = Semicircle(I', I)
d = Semicircle(B', B)
f = Line(O, I)
g = OrthogonalLine(O, f)
C = Intersect(c, g, 1)
h = Line(C, A)
D = Intersect(d, g, 1)
i = Line(D, h)
E = Intersect(f, i)

Tool: OpDivisión
Elige O e I, y luego el primer punto y el segundo punto
Input: O, I, A, B
Output: E

f = Line(O, I)
g = OrthogonalLine(O, f)
I' = Mirror(I, O)
c = Semicircle(I', I)
Binv = Mirror(B, c)
Binv' = Mirror(Binv, O)
e = Semicircle(Binv', Binv)
F = Intersect(e, g, 1)
C = Intersect(c, g, 1)
h = Line(C, A)
i = Line(F, h)
E = Intersect(f, i)