Analytic geometry

[yangyongkang]

I will extend this question. Calculate the area enclosed by the three tangent lines of the circle.

Code:

(det([[l1(1,1),l1(1,2),1],[l2(1,1),l2(1,2),1],[l3(1,1),l3(1,2),1]])))

Get the answer

Code:

(-2*tan(alpha/2)^2*tan(beta/2)+2*tan(alpha/2)^2*tan(gamma/2)-2*tan(beta/2)^2*tan(gamma/2)+2*tan(alpha/2)*tan(beta/2)^2-2*tan(alpha/2)*tan(gamma/2)^2+2*tan(beta/2)*tan(gamma/2)^2)/(tan(alpha/2)*tan(beta/2)+tan(alpha/2)*tan(gamma/2)+tan(beta/2)*tan(gamma/2)+tan(alpha/2)^2*tan(beta/2)^2*tan(gamma/2)^2+tan(alpha/2)^2*tan(beta/2)*tan(gamma/2)+tan(alpha/2)*tan(beta/2)^2*tan(gamma/2)+tan(alpha/2)*tan(beta/2)*tan(gamma/2)^2+1)

Defining function

Code:

f(alpha,beta,gamma):=(-2*tan(alpha/2)^2*tan(beta/2)+2*tan(alpha/2)^2*tan(gamma/2)-2*tan(beta/2)^2*tan(gamma/2)+2*tan(alpha/2)*tan(beta/2)^2-2*tan(alpha/2)*tan(gamma/2)^2+2*tan(beta/2)*tan(gamma/2)^2)/(tan(alpha/2)*tan(beta/2)+tan(alpha/2)*tan(gamma/2)+tan(beta/2)*tan(gamma/2)+tan(alpha/2)^2*tan(beta/2)^2*tan(gamma/2)^2+tan(alpha/2)^2*tan(beta/2)*tan(gamma/2)+tan(alpha/2)*tan(beta/2)^2*tan(gamma/2)+tan(alpha/2)*tan(beta/2)*tan(gamma/2)^2+1)

Deriving a trace to track its suspicious extreme point

Code:

solve([diff(f(x,y,z),x)=0,diff(f(x,y,z),y)=0,diff(f(x,y,z),z)=0],[x,y,z])

XCAS calculates the result

Code:

[[pi/3,pi/3,pi/3],[-pi/3,-pi/3,-pi/3],[2*atan(y),-2.33112237041,2*atan(y)],[2*atan(y),-1.49890655757e-26,2*atan(y)],[2*atan(y),2.33112237041,2*atan(y)],[-2.33112237041,2*atan((x-(sqrt(3)))/(sqrt(3)*x+1)),2*atan((-x-(sqrt(3)))/(sqrt(3)*x-1))],[-1.49890655757e-26,2*atan((x-(sqrt(3)))/(sqrt(3)*x+1)),2*atan((-x-(sqrt(3)))/(sqrt(3)*x-1))],[2.33112237041,2*atan((x-(sqrt(3)))/(sqrt(3)*x+1)),2*atan((-x-(sqrt(3)))/(sqrt(3)*x-1))],[-2.33112237041,2*atan((-x-(sqrt(3)))/(sqrt(3)*x-1)),2*atan((x-(sqrt(3)))/(sqrt(3)*x+1))],[-1.49890655757e-26,2*atan((-x-(sqrt(3)))/(sqrt(3)*x-1)),2*atan((x-(sqrt(3)))/(sqrt(3)*x+1))],[2.33112237041,2*atan((-x-(sqrt(3)))/(sqrt(3)*x-1)),2*atan((x-(sqrt(3)))/(sqrt(3)*x+1))]]

Is there any other good solution?
Thank you