Position an axis at a point location

I have a plot with a line graph. I need to place a new axis at a point/coordinate and plot a pie chart on the new axis so the pie is centered on the point.

import matplotlib.pyplot as plt
import numpy as np
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(6, 3))
ax.set_xlim(0,16)
ax.set_ylim(8,12)

#Plot a sine wave
x = np.arange(0, 5*np.pi, 0.1)
y = np.sin(x)+10
ax.plot(x, y, color="blue")

#Plot a red point at x=12, y=10
ax.plot(12,10,marker="o", color="red")

#Add a new axes to the plot

#Normalize the points coordinates to range between 0 and 1
x_norm = (12-0)/(16-0) #0.75
y_norm = (10-8)/(12-8) #0.5

#Add an ax at the normalized coordinates
left=0.75
bottom=0.5
width=0.1
height=0.1
sub_ax = fig.add_axes(rect=(left, bottom, width, height))

sub_ax.pie((0.2,0.3,0.5))

The pie is centered on the new axis center. I cant figure out the logic to get it centered at the point?

You could use the inset_axes method, (based on the last option from this answer) e.g.,

import matplotlib.pyplot as plt
import numpy as np
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(6, 3))
ax.set_xlim(0,16)
ax.set_ylim(8,12)

#Plot a sine wave
x = np.arange(0, 5*np.pi, 0.1)
y = np.sin(x)+10
ax.plot(x, y, color="blue")

#Plot a red point at x=12, y=10
point = (12, 10)
ax.plot(*point, marker="o", color="red")

# width and height in data coordinates
width = 2
height = 0.5

# subtract half the width/height from the left/bottom positions
position = [point[0] - width / 2, point[1] - height / 2, width, height]

sub_ax = ax.inset_axes(position, transform=ax.transData)
sub_ax.pie((0.2,0.3,0.5))