空間図形「前橋工科大学2020年第2問」をsympyでやってみた。

作図はありません

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オリジナル

上と同じです。大学入試数学問題集成＞

sympyで(positiveで)

from sympy import *
a=Symbol('a', positive=True)
b=Symbol('b', positive=True)
c=Symbol('c', positive=True)
O=Point(0,0,0)
A1,B1,C1=map(Point,([  sqrt(a),0,0],[0,  sqrt(b),0],[0,0,  sqrt(c)]))
A2,B2,C2=map(Point,([1/sqrt(a),0,0],[0,1/sqrt(b),0],[0,0,1/sqrt(c)]))
H1=Plane(A1,B1,C1).projection(O)
H2=Plane(A2,B2,C2).projection(O)
print("#(1)",H1)
print("#(2)",H2)
print("#(3)",(H1-O).dot((H2-O))/(O.distance(H1)*O.distance(H2))           )
print("#(3)",(H1-O).dot((H2-O))/(O.distance(H1)*O.distance(H2)).simplify())
#(1) Point3D(sqrt(a)*b*c/(a*b + a*c + b*c), a*sqrt(b)*c/(a*b + a*c + b*c), a*b*sqrt(c)/(a*b + a*c + b*c))
#(2) Point3D(sqrt(a)/(a + b + c), sqrt(b)/(a + b + c), sqrt(c)/(a + b + c))
#(3) 3*a*b*c/((a + b + c)*(a*b + a*c + b*c)*sqrt(a/(a + b + c)**2 + b/(a + b + c)**2 + c/(a + b + c)**2)*sqrt(a**2*b**2*c/(a*b + a*c + b*c)**2 + a**2*b*c**2/(a*b + a*c + b*c)**2 + a*b**2*c**2/(a*b + a*c + b*c)**2))
#(3) 3*sqrt(a)*sqrt(b)*sqrt(c)/(sqrt(a + b + c)*sqrt(a*b + a*c + b*c))

sympyで(varで)

from sympy import *
var('a b c')
O=Point(0,0,0)
A1,B1,C1=map(Point,([  sqrt(a),0,0],[0,  sqrt(b),0],[0,0,  sqrt(c)]))
A2,B2,C2=map(Point,([1/sqrt(a),0,0],[0,1/sqrt(b),0],[0,0,1/sqrt(c)]))
H1=Plane(A1,B1,C1).projection(O)
H2=Plane(A2,B2,C2).projection(O)
print("#(1)",H1)
print("#(2)",H2)
print("#(3)",(H1-O).dot((H2-O))/(O.distance(H1)*O.distance(H2))           )
print("#(3)",(H1-O).dot((H2-O))/(O.distance(H1)*O.distance(H2)).simplify())
#(1) Point3D(sqrt(a)*b*c/(a*b + a*c + b*c), a*sqrt(b)*c/(a*b + a*c + b*c), a*b*sqrt(c)/(a*b + a*c + b*c))
#(2) Point3D(sqrt(a)/(a + b + c), sqrt(b)/(a + b + c), sqrt(c)/(a + b + c))
#(3) 3*a*b*c/((a + b + c)*(a*b + a*c + b*c)*sqrt(a/(a + b + c)**2 + b/(a + b + c)**2 + c/(a + b + c)**2)*sqrt(a**2*b**2*c/(a*b + a*c + b*c)**2 + a**2*b*c**2/(a*b + a*c + b*c)**2 + a*b**2*c**2/(a*b + a*c + b*c)**2))
#(3) 3*a*b*c/(sqrt(a*b*c/(a*b + a*c + b*c))*(a + b + c)*(a*b + a*c + b*c)*sqrt(1/(a + b + c)))