Browse · MathNet
PrintEstonian Mathematical Olympiad
Estonia geometry
Problem
A point is chosen on the side of a rectangle (). The line segments and intersect at point . Among the triangles , , , and , there are exactly two pairs of triangles with equal area (the order of components in a pair is not taken into account). Find the ratio of the lengths of the line segments and .
Solution
Denote the area of a figure by . As we have (Fig. 39). Hence (, ) is one pair of triangles with equal area regardless of the choice of point .
In order to have exactly two such pairs, none of the remaining three triangles can have the same area as triangles DEF and BCF. Thus the second pair must come from among triangles ADE, BEF and DCF. On the other hand, implying . Hence each of triangles ADE and BEF has an area smaller than that of the triangle DCF. Thus the second pair of triangles with equal area is (ADE, BEF). Taking into account the equality , we obtain .
As and , triangles BEF and DCF are similar. Hence , implying
In order to have exactly two such pairs, none of the remaining three triangles can have the same area as triangles DEF and BCF. Thus the second pair must come from among triangles ADE, BEF and DCF. On the other hand, implying . Hence each of triangles ADE and BEF has an area smaller than that of the triangle DCF. Thus the second pair of triangles with equal area is (ADE, BEF). Taking into account the equality , we obtain .
As and , triangles BEF and DCF are similar. Hence , implying
Final answer
sqrt(2)/2
Techniques
Angle chasingQuadrilaterals