Analytic geometry

[yangyongkang]

Hello everyone. I recently encountered a planar geometry problem, I tried to solve it with algebraic methods.This requires me to solve a series of equations.

Code:

f := proc (alpha) options operator, arrow, function_assign; x*cos(alpha)+y*sin(alpha) end proc;l1:=solve([f(alpha) = 1, f(beta) = 1], [x, y]);l2:=solve([f(alpha) = 1, f(gamma) = 1], [x, y]);l3:=solve([f(beta) = 1, f(gamma) = 1], [x, y]);solve([(l1(1,1)-l2(1,1))^2+(l1(1,2)-l2(1,2))^2=(l1(1,1)-l3(1,1))^2+(l1(1,2)-l3(1,2))^2,(l1(1,1)-l2(1,1))^2+(l1(1,2)-l2(1,2))^2=(l3(1,1)-l2(1,1))^2+(l3(1,2)-l2(1,2))^2],[beta, alpha])

XCAS can't solve it
Wolfram Mathematica 11.3 can't solve it


Wolfram Mathematic 11.3 code

Code:

f[m_] := x*Cos[m] + y*Sin[m]; l1 := {x, y} /. 
  Solve[f[t1] == 1 && f[t2] == 1, {x, y}]; l2 := {x, y} /. 
  Solve[f[t1] == 1 && f[t3] == 1, {x, y}]; l3 := {x, y} /. 
  Solve[f[t2] == 1 && f[t3] == 1, {x, 
    y}]; Solve[(l1[[1, 1]] - l2[[1, 1]])^2 + (l1[[1, 2]] - 
       l2[[1, 2]])^2 == (l1[[1, 1]] - l3[[1, 1]])^2 + (l1[[1, 2]] - 
       l3[[1, 2]])^2 && (l3[[1, 1]] - l2[[1, 1]])^2 + (l3[[1, 2]] - 
       l2[[1, 2]])^2 == (l1[[1, 1]] - l2[[1, 1]])^2 + (l1[[1, 2]] - 
       l2[[1, 2]])^2, {t2, t1}]

But Maple2018 seems to solve

Code:

f := proc (alpha) options operator, arrow, function_assign; x*cos(alpha)+y*sin(alpha) end proc; l1 := subs(solve([f(alpha) = 1, f(beta) = 1], [x, y])[1], [x, y]); l2 := subs(solve([f(alpha) = 1, f(gamma) = 1], [x, y])[1], [x, y]); l3 := subs(solve([f(beta) = 1, f(gamma) = 1], [x, y])[1], [x, y]); allvalues(solve([(l1[1]-l2[1])^2+(l1[2]-l2[2])^2 = (l1[1]-l3[1])^2+(l1[2]-l3[2])^2, (l1[1]-l2[1])^2+(l1[2]-l2[2])^2 = (l2[1]-l3[1])^2+(l2[2]-l3[2])^2], [beta, alpha]))

Its geometric meaning is an equilateral triangle surrounded by three tangents of a circle.
The equation of the circle is

Code:

x^2+y^2=1

Randomly take three parameters and draw this image

Code:

plotimplicit(x*cos(1)+y*sin(1)=1);plotimplicit(x*cos(-8)+y*sin(-8)=1);plotimplicit(x*cos(3)+y*sin(3)=1);plotimplicit(x^2+y^2=1)

Sure enough, the three tangent lines

Sorry my poor english