OM = 1

$P'R^2=P'H^2+HR^2$
=$OP'^2-OH^2+(OH-OR)^2$
=$OP'^2-OH^2+OH^2+OR^2-2\times OH \times OR$
=$OP'^2+OR^2-2\times OH \times OR$
=$OP'^2+OR^2-2 \times OR \times OP' \times cos(a)$

    trigonométrie
cosinus d'un angle

théorème d'Al Kashi

$cos(a)=\frac{OH}{OP'}$
donc $OH = OP' \times cos(a)$

O: (0, 0)
I: (1, 0)
angle = 0.5235987755982988 rad (slider from 0 to 1.5707963267948966, step 1°)
c = Circle(O, I)
M: (cos(angle), sin(angle))
M': (x(M), 0)
f = Ray(O, M)
P = Point(f)
P': (x(P), 0)
g = Segment(M, M')
PP' = Segment(P, P')
i = Ray(O, M')
a = Angle(I, O, M)
b: x(M')
N: 1
OM = Segment(O, M)
OP = Segment(O, P)
OM' = Segment(O, M')
OP' = Segment(O, P')
texte3: "OM' = " + OM' + ""
texte2: "\frac{OM'}{OM}=" + (LaTeX[b]) + ""
texte4: "OP = " + OP + ""
texte5: "OP' = " + OP' + ""
texte6: "\frac{OP'}{OP}=" + (LaTeX[b]) + ""
α = Angle(P', M', M)
β = Angle(P, P', M')
R = Point(OP)
OR = Segment(O, R)
RP' = Segment(R, P')
γ = Angle(P, P', O)
t1, r, o, p' = Polygon(O, P', R)
texte8: "cos(a)=\frac{OM'}{OM}=\frac{OP'}{OP}=" + (LaTeX[b]) + ""
h = OrthogonalLine(P', OP)
H = Intersect(OP, h)
δ = Angle(P', H, P)
j = Segment(P', H)