ABC and XYZ are similar triangles and the circumcircle of the triangle XYZ is the incircle of the triangle ABC. If k = Area of ABC/Area of XYZ, then find the minimum value of k.

(In reply to mathematica solution by Daniel)

There must be something wrong here. Since triangle ABC's incircle is the circumcircle of triangle XYZ, triangle ABC must be larger than triangle XYZ, so k must be greater than 1.

Also, playing around with Geometer's Sketchpad, constructing a triangle's circumcircle and then, by placing tangents to the circumcircle parallel to the original triangle's sides, I get a ratio of areas at least 4.  When the triangle is equilateral is when I get the area of 4, which seems to be the minimum sought.

With the two triangles being equilateral, when perpendicular bisectors are drawn from the sides (serving both when their corresponding sides are parallel), six right triangles are formed in each equilateral triangle. The hypotenuse is twice the side of the shorter leg, thereby giving the radius of the smaller triangle's circumcircle a radius of twice this shorter leg's length. But this is the shorter leg of the larger size right triangle, 1/6 of the larger equilateral triangle, and so the ratio of their linear dimensions is 2, and of their areas is 4.  As mentioned previously, when the triangles are not equilateral, the ratio is larger in GSP.

Posted by Charlie on 2009-05-26 13:21:07