Given a regular hexagon, a point inside is chosen and lines to each vertex of the hexagon are drawn, dividing it into six triangles. If the triangles are alternately shaded grey and white, show that the area of the grey triangles is the same as the area of the white triangles.
, 2011
Solution
Expand the hexagon to make a big equilateral triangle, and drop perpendiculars from the point inside to the sides of the triangle. The area of the unshaded region of the hexagon, which is three triangles of equal bases, is equal to side length multiplied by the sum of the heights. But in an equilateral triangle, no matter where the interior point is chosen, the sum of these dropped perpendiculars is constant (equal to the height of the triangle). So the unshaded and the shaded regions have equal area.
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.