$\textsf{\LARGE O}$

$\textsf{\LARGE I}$

$\textsf{\LARGE A}$

Addition

We will call sA the conic of semi-axis s and foci O (fixed) and A.
The sum of the conics sA and sB is the conic s(A+B).

Multiplication

The product of the conics sA and sB is the conic s(A×B).
The conic sI is the neutral element.

Division

$\textsf{\LARGE B}$

Subtraction

The subtraction of the conics sA and sB is the conic s(A-B).

$\textsf{\LARGE A−B}$

$\textsf{\LARGE A'}$

B−A

A⁻¹

The division of the conics sA and sB is the conic s(A/B)

B/A

$\textsf{\LARGE A+B}$

$\textsf{\LARGE A×B}$

$\textsf{\LARGE A/B}$

The neutral element is the circle of diameter s (degenerate ellipse sO)

The opposite element of the conic sA is the conic sA'

The inverse element of the conic sA is the conic sA⁻¹

$\textsf{\LARGE A⁻¹}$

Equation

(p,0)

If we make the origin of coordinates coincide with O and the point (1,0) with I, the point P 
will correspond to (p,0), so we can represent the conic sP with the corresponding equation:

\left(\frac{2x-p}{s}\right)^2-\frac{4y^2}{p^2-s^2}=1

s

s: 3
O: (-1.57, 1.72)
c = Circle(O, s)
I: (0, 1.5)
r = Line(O, I)
A = Point(r)
I' = Mirror(I, O)
op1 = true
op3 = false
op4 = false
B = Point(r)
op2 = false
AmenosB = OpResta(O, A, B)
A' = Mirror(A, O)
verBmenosA = false
BmenosA = OpSuma(O, B, A')
verBporA = false
e_1 = Semicircle(A', A)
k = Circle(O, I)
Ainv = Mirror(A, k)
verAinv = false
BentreA = OpDivisión(O, I, B, A)
verBentreA = false
AmásB = OpSuma(O, A, B)
AporB = OpProducto(O, I, A, B)
AentreB = OpDivisión(O, I, A, B)
C_1 = OpDivisión(O, I, I, A)
D = Point(c)
pp = Ellipse(O, A, s / 2)
q = Ellipse(O, B, s / 2)
s_1 = Ellipse(O, AmásB, s / 2)
j_1 = Ellipse(O, AmenosB, s / 2)
a = PathParameter(A)
f = Ellipse(O, AporB, s / 2)
g = Ellipse(O, AentreB, s / 2)
op5 = false
O_0: (-1.57, 1.72)
I_0: (0, 1.5)
P = Point(r)
anima = 0.661 (slider from 0 to 1)
d = Ellipse(O, P, s / 2)
p: x(P)
G = Point(c)
l = Segment(O, G)

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)