In Quad.PQRS,
Opposite sides PQ = SR 
                           PS = QR

∴The figure obtained by joining the mid points of sides 
of a quadrialteral is a parallelogram

opposite sides

opposite angles

sum of angles

conclusion

Length = 15 (slider from 6 to 16, step 1)
A = (-17.975630655535586, 5.550844403958454)
B = Point(Circle[A, Length])
f = Segment(A, B)
C = (-5.568397555740401, 14.900580775589928)
D = (-16.247480330921253, 17.51496203590394)
q1, a_1, b, c, d = Polygon(A, B, C, D)
g = Segment(A, C)
h = LineBisector(a_1)
P = Intersect(h, a_1)
i = LineBisector(b)
Q = Intersect(i, b)
j = LineBisector(c)
R = Intersect(c, j)
k = LineBisector(d)
S = Intersect(k, d)
l = Segment(P, Q)
m = Segment(Q, R)
n = Segment(R, S)
p = Segment(S, P)
α = Angle(Q, P, S)
β = Angle(S, R, Q)
γ = Angle(R, Q, P)
δ = Angle(P, S, R)
sumofadjacentangles: α + γ
text2: "Sum of adjacent angles 
∠P + ∠S  = " + α + " +" + γ + "=" + sumofadjacentangles + " "
a = false
e = false
o = false
q = false
text1: "Opposite angles
∠ P = " + β + " ∠Q =" + γ + "
∠R =" + α + "   ∠S = " + δ + "
So, ∠P = ∠R & ∠Q =∠R"