1954 AHSME Problems/Problem 46
In the diagram, if points and are points of tangency, then equals:
First we extend the line with and the line with so that they both meet the line with , forming an equilateral triangle. Let the vertices of this triangle be , , and . We know it is equilateral because of the angle of shown, and because the tangent lines and are congruent. We can see, because , , and are points of tangency, that circle is inscribed in . The height of an equilateral triangle is exactly times the radius of a circle inscribed in it. Let the height of be . We can see that the radius of the circle equals . Thus Subtracting from gives us so our answer is .