An equilateral triangle and regular hexagon are inscribed in the same circle. Find the ratio of the area of the triangle to that of the hexagon.

Source: mathcontest.olemiss.edu 12/7/2009

**SOLUTION
**First, rotate the equilateral triangle around center of the circle so that vertex coincides with a vertex of the regular hexagon. Draw segment .

**Show that triangle ** **is congruent to triangle **

Triangle is congruent to triangle by ASA (angle side angle):

(see why below)

reflexive property

(see why below)

Since point is equidistant from the endpoints of segment , is on the the perpendicular bisector of . Furthermore, we can prove that is also the angle bisector of by proving that right triangle is congruent to right triangle .

Since (center angle intercepting the same arc as the inscribed angle of an equilateral triangle), half of it is equal to . Thus, and .

Similarly, we can prove that and .

**Unit area**

Let the area of triangle be the unit area. Then, we can visually determine that the equilateral triangle is made up of 3 units and the regular hexagon is made up of 6 units.

Thus, the ratio of the area of the equilateral triangle to the area of the regular hexagon is

.

**Answer**: .

Being a complete newbie, all I can say is thanks for sharing this.