Problem
GEO-B1-M02-P026 Four Congruent Triangles
Can four pairwise congruent triangles be assembled into a convex quadrilateral with no parallel sides? If yes, describe a construction and prove that it works.
C. Hint 1. Start not with four triangles, but with a kite made from two congruent isosceles triangles.
D. Hint 2. Cut each of those two isosceles triangles by its altitude to the base.
E. Full solution.
Take an isosceles triangle that is not equilateral, and glue two copies along a lateral side so that a convex kite is formed. Its adjacent sides are equal in pairs, but opposite sides need not be parallel.
Choose the original isosceles triangle so that its vertex angle is not equal to its base angle. Then in the resulting kite no pair of opposite sides is parallel: otherwise equality of corresponding angles with a transversal would force the vertex angle to equal a base angle.
Now draw the altitude to the base in each of the two isosceles triangles. In an isosceles triangle this altitude splits it into two congruent right triangles. The two larger isosceles triangles were congruent, so all four small right triangles are pairwise congruent.
Thus the same kite is assembled from four congruent triangles and has no parallel sides. Therefore the answer is yes.
A. Source analysis. Main objects: four congruent triangles, a convex quadrilateral, and absence of parallel sides. The obvious approach is to try to prove that parallel sides are forced, but the hidden move is to build a counterexample through a kite. The needed transformation is to first assemble two larger isosceles parts and then cut them into four congruent pieces. Number of key ideas: 2.
F. Difficulty justification. This is Level 6: a regional-style construction counterexample without heavy machinery.
G. Check. This is not a one-step exercise: one must invent an intermediate figure and verify both the congruence of the small triangles and the absence of parallel sides.