Je\ dán\ pravidelný\ čtyřboký\ jehlan,\ ve\ kterém\ je
\a=\left|AB  \right|, 2a=\left|AV  \right| .\ Určete\ odchylku\ přímky\ BD\ 
\od\ roviny\ CDV. 
 

Zadání

\left| DS \right| =\sqrt{(\frac{a}{2})^2+(\frac{a}{2})^2}=\frac{a\sqrt{2}}{2}

sinα=\frac{\left| DS \right| }{\left| PS \right| }=\frac{\frac{\sqrt{2}a}{2}}{\frac{\sqrt{7}a}{\sqrt{15}\sqrt{2}}}=\sqrt{\frac{7}{15}}

a = 2.9 (slider from 0.5 to 20, step 0.2)
A = (-1.06, -2.82)
B = Point(Circle[A, a])
f = Segment(A, B)
mnohoúhelník1, g, h, i, j, C, D = Polygon(A, B, 4)
k = Segment(D, B)
l = Segment(A, C)
S = Intersect(l, k)
m = OrthogonalLine(S, xOyPlane)
b = Sphere(S, (a * (7 / 2)^(1 / 2)))
E, V = Intersect(m, b)
c, stenaABCD, stenaABV, stenaBCV, stenaCDV, stenaADV, hranaAB, hranaBC, hranaCD, hranaAD, hranaAV, hranaBV, hranaCV, hranaDV = Pyramid(A, B, C, D, V)
d = Plane(C, D, V)
n = Line(D, B)
p = OrthogonalLine(V, i, d)
S_{AB} = Midpoint(A, B)
S_{DC} = Midpoint(D, C)
q = Segment(S_{AB}, S_{DC})
e = Plane(S_{AB}, S_{DC}, V)
r = Segment(V, S_{DC})
s = Segment(V, S_{AB})
t = Segment(V, S)
f_1 = OrthogonalLine(S, r, e)
P = Intersect(p, f_1)
α = Angle(S, P, S_{DC}, e)
g_1 = Line(P, D)
o = Plane(B, P, S)
β = Angle(B, D, P, o)
u = true
γ = Angle(S, V, P, e)
text5: "Nejprve\ spočti\ úhel\ SVS_{DC}:
\\sinα=\frac{SS_{DC}}{VS_{DC}}=\frac{1}{\sqrt{15}}, \; α= " + (LaTeX[γ]) + "
\\Dále\ spočtem\ v: v^2=(\left|VS_{DC}  \right| )^2-(SS_{DC})^2
\\v=\sqrt{\frac{7}{2}}a
\\Následovně spočtem \left| SP \right| 
\\sinα =\frac{\left| SP \right| }{v}=\frac{1}{\sqrt{15}} \;  \; ,\left| SP \right| =\frac{\sqrt{7}}{\sqrt{15}\sqrt{2}}"