【面】三角形的外接圆和内切圆

更新于
三角形外接圆的圆心是三角形垂直平分线的交点。三角形内切圆的圆心是三角形角平分线的交点。
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import numpy as np
from scipy.linalg import solve
import matplotlib.pyplot as plt
from matplotlib.patches import Circle

# 画三角形
def plot_triangle(A, B, C):
    x = [A[0], B[0], C[0], A[0]]
    y = [A[1], B[1], C[1], A[1]]
    ax = plt.gca()
    ax.plot(x, y, linewidth=2)

# 画圆
def draw_circle(x, y, r):
    ax = plt.gca()
    cir = Circle(xy=(x, y), radius=r, alpha=0.5)
    ax.add_patch(cir)
    ax.plot()

# 三角形的外接圆
def get_outer_circle(A, B, C):
    xa, ya = A[0], A[1]
    xb, yb = B[0], B[1]
    xc, yc = C[0], C[1]

    # 两条边的中点
    x1, y1 = (xa + xb) / 2.0, (ya + yb) / 2.0
    x2, y2 = (xb + xc) / 2.0, (yb + yc) / 2.0

    # 两条线的斜率
    ka = (yb - ya) / (xb - xa) if xb != xa else None
    kb = (yc - yb) / (xc - xb) if xc != xb else None

    alpha = np.arctan(ka) if ka != None else np.pi / 2
    beta = np.arctan(kb) if kb != None else np.pi / 2

    # 两条垂直平分线的斜率
    k1 = np.tan(alpha + np.pi / 2)
    k2 = np.tan(beta + np.pi / 2)

    # 圆心
    y, x = solve([[1.0, -k1], [1.0, -k2]], [y1 - k1 * x1, y2 - k2 * x2])
    # 半径
    r1 = np.sqrt((x - xa)**2 + (y - ya)**2)

    return(x, y, r1)

# 三角形的内切圆
def get_inner_circle(A, B, C):
    xa, ya = A[0], A[1]
    xb, yb = B[0], B[1]
    xc, yc = C[0], C[1]

    ka = (yb - ya) / (xb - xa) if xb != xa else None
    kb = (yc - yb) / (xc - xb) if xc != xb else None

    alpha = np.arctan(ka) if ka != None else np.pi / 2
    beta = np.arctan(kb) if kb != None else np.pi / 2

    a = np.sqrt((xb - xc)**2 + (yb - yc)**2)
    b = np.sqrt((xa - xc)**2 + (ya - yc)**2)
    c = np.sqrt((xa - xb)**2 + (ya - yb)**2)

    ang_a = np.arccos((b**2 + c**2 - a**2) / (2 * b * c))
    ang_b = np.arccos((a**2 + c**2 - b**2) / (2 * a * c))

    # 两条角平分线的斜率
    k1 = np.tan(alpha + ang_a / 2)
    k2 = np.tan(beta + ang_b / 2)
    kv = np.tan(alpha + np.pi / 2)

    # 求圆心
    y, x = solve([[1.0, -k1], [1.0, -k2]], [ya - k1 * xa, yb - k2 * xb])
    ym, xm = solve([[1.0, -ka], [1.0, -kv]], [ya - ka * xa, y - kv * x])
    r1 = np.sqrt((x - xm)**2 + (y - ym)**2)

    return(x, y, r1)
 
if __name__ == '__main__':
    A = (1., 1.)
    B = (5., 2.)
    C = (5., 5.)

    plt.axis('equal')
    plt.axis('off')
    plot_triangle(A, B, C)

    x, y, r1 = get_outer_circle(A, B, C)
    plt.plot(x, y, 'ro')
    draw_circle(x, y, r1)

    x_inner, y_inner, r_inner = get_inner_circle(A, B, C)
    plt.plot(x_inner, y_inner, 'ro')
    draw_circle(x_inner, y_inner, r_inner)

    plt.show()
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