a triangle is formed by drawing tangents at A,B,C to the circumcircle of triangle ABC prove that the perimeter of this triangle is 2RtanAtanBtanC where R is the radius of circumcircle.
See the diagram.
This shows 1/3 of the proof.
Consider the part of the external triangle opposite angle A. Call angle A theta.
The diagram shows that the length for the part of the triangle opposite angle A is
2*R*tan(theta) = 2*R*tan(A)
Note: The 2*theta is measured from the center of the circle.
Repeating the procedure for the parts opposite B and C and summing, we get:
Perimeter = 2*R*tan(A) + 2*R*tan(B) + 2*R*tan(C) = 2R(tan(A) + tan(B) + tan(C))
Is this the formula you want to prove?
Hello, snowtea!
Lovely work!
You found that: .
. . which involves the sum of the tangents.
The original equation has the product of the tangents.
But not to worry . . . Here's a surprising theorem:
. . In
Proof
Take tangents: .
. . . . . . . . .
. . . . . . . . . .
. . . .