Equation of a curve through three points

This is an example of Lagrange Polynomial through 3 points

(a, A)

(b, B)

Projections

Curve 1 (a, A)

Curve 2 (b, B)

Curve 1-2-3

(c, C)

Curve 3 (c, C)

A = (3.4973081997225157, 9.16652877310255)
B = (14, -4)
x_A: x(A)
y_A: y(A)
x_B: x(B)
y_B: y(B)
c_1 = OrthogonalLine(A, xAxis)
d = OrthogonalLine(B, xAxis)
C_1 = Intersect(c_1, xAxis)
D = Intersect(d, xAxis)
e = Segment(A, C_1)
f = Segment(B, D)
g = false
m_A: y_A / (x_A - x_B)
c_A: -y_A * x_B / (x_A - x_B)
m_B: y_B / (x_B - x_A)
c_B: -y_B * x_A / (x_B - x_A)
m_{AB}: m_A + m_B
c_{AB}: c_A + c_B
i = false
j = false
k = false
C = (24.641658817116408, 2.2473408678171514)
x_C: x(C)
y_C: y(C)
a: y = y_A * (x - x_B) * (x - x_C) / ((x_A - x_B) * (x_A - x_C))
b: y = y_B * (x - x_C) * (x - x_A) / ((x_B - x_C) * (x_B - x_A))
c: y = y_C * (x - x_A) * (x - x_B) / ((x_C - x_A) * (x_C - x_B))
l = OrthogonalLine(C, xAxis)
E = Intersect(l, xAxis)
m = Segment(C, E)
text1: "" + (LaTeX[a]) + " = A \frac{\left(  x - b \right)\left( x - c  \right) }{\left( a - b  \right) \left(  a - c \right) }"
text2: "" + (LaTeX[b]) + "  = B \frac{\left(  x - c \right)\left( x - a  \right) }{\left( b - c  \right) \left(  b - a \right) }"
text4: "" + (LaTeX[c]) + "   = C \frac{\left(  x - a \right)\left( x - b  \right) }{\left( c - a  \right) \left(  c - b \right) }"
n = false
h: y = y_A * (x - x_B) * (x - x_C) / ((x_A - x_B) * (x_A - x_C)) + y_B * (x - x_C) * (x - x_A) / ((x_B - x_C) * (x_B - x_A)) + y_C * (x - x_A) * (x - x_B) / ((x_C - x_A) * (x_C - x_B))
text3: "" + (LaTeX[h]) + " = A \frac{\left(  x - b \right)\left( x - c  \right) }{\left( a - b  \right) \left(  a - c \right) } + B \frac{\left(  x - c \right)\left( x - a  \right) }{\left( b - c  \right) \left(  b - a \right) } + C \frac{\left(  x - a \right)\left( x - b  \right) }{\left( c - a  \right) \left(  c - b \right) }"