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.

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- Dec 12th 2010, 07:59 AMprasumtrigo
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.

- Dec 12th 2010, 08:53 AMsnowtea
See the diagram.

Attachment 20070

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? - Dec 12th 2010, 10:21 AMSoroban
Hello, snowtea!

Lovely work!

You found that: .

. . which involves theof the tangents.*sum*

The original equation has theof the tangents.*product*

But not to worry . . . Here's a surprising theorem:

. . In

**Proof**

Take tangents: .

. . . . . . . . .

. . . . . . . . . .

. . . .

- Dec 13th 2010, 06:48 AMprasum