Wendel’s theorem in the plane

Solution. Step 1: geometric reformulation. A triangle inscribed in a circle contains the centre $O$ iff the three arcs between consecutive vertices (measured along the circle) are all shorter than a semicircle. Equivalently, the triangle fails to contain $O$ iff the three points lie in some semicircle. (The reason: a triangle with vertices on a circle contains the centre iff no diameter separates the points from the centre, which is the same as saying no open semicircle contains all three.)

Step 2: parametrise. By rotational symmetry, condition on $P_1$ being at angle $0$ on the unit circle. Let $X, Y \in [0,1]$ be the angles of $P_2, P_3$ measured as a fraction of the full circle (so angle $= 2\pi X$ and $2\pi Y$); both are independently uniform on $[0,1]$.

Given $0 < X < Y < 1$ (the event $X < Y$, probability $\tfrac12$), the three arcs as fractions of the circle are $X$, $Y - X$, and $1 - Y$, summing to $1$. All three are strictly below $\tfrac12$ iff $$X < \tfrac12, \qquad Y - X < \tfrac12, \qquad 1 - Y < \tfrac12 \;\;\Longleftrightarrow\;\; Y > \tfrac12.$$

Step 3: compute the area. In the triangle $\{(X, Y) : 0 < X < Y < 1\}$, the region satisfying the three conditions above is itself a triangle with vertices $(0, \tfrac12)$, $(\tfrac12, \tfrac12)$, $(\tfrac12, 1)$, of area $\tfrac12 \cdot \tfrac12 \cdot \tfrac12 = \tfrac18$.

Hence $\mathbb{P}(\text{triangle contains } O \;\mathrm{and}\; X < Y) = \tfrac18$. By symmetry (interchanging $X$ and $Y$), the unconditional probability is $$\boxed{\mathbb{P} = 2 \cdot \tfrac18 = \tfrac14.} \qedhere$$

import numpy as np
# 3 uniform points on circle; triangle contains origin iff the signed areas
# (origin with each edge) all share the sign of the full triangle's signed area.
theta = np.random.rand(3, 10**6) * 2*np.pi
x, y = np.cos(theta), np.sin(theta)
s_tri = (x[1]-x[0])*(y[2]-y[0]) - (x[2]-x[0])*(y[1]-y[0])
s01 = x[0]*y[1] - x[1]*y[0]
s12 = x[1]*y[2] - x[2]*y[1]
s20 = x[2]*y[0] - x[0]*y[2]
contains = (np.sign(s_tri)==np.sign(s01)) & \
           (np.sign(s_tri)==np.sign(s12)) & \
           (np.sign(s_tri)==np.sign(s20))
print(f"sim: {contains.mean():.5f}   exact: 0.25")
# sim: 0.25033   exact: 0.25