How many congruent triangles are formed by connecting the midpoints of the three sides of a scalene triangle?
4. Each midpoint connector is parallel to the opposite side. Therefore each of the small three triangles created by a midpoint connector and the sides of the original triangle are similar. But they are all also exactly half the linear dimensions of the original triangle, therefore congruent to each other, with area equal to 1/4 the area of the original triangle. The triangle in the middle is also similar, because all its sides are parallel to the sides of the original, hence the angles are the same (although it is rotated 180°). Since the other 3 together are 3/4 the area of the original, this one is 1/4 the area of the original, too, and all 4 are congruent. (Since I didn’t use the fact that original triangle is scalene, to be a rigorous proof it would need some tightening, but that’s not what was asked for.
