1020.15 – A Triangle with Given Midpoints
Given three, non-colinear points, the problem is to find method whereby a triangle can be constructed such that the midpoints of its sides are the given points. Explain why your method works.
Just drawing a single triangle and its midpoints will suggest the solution, which is based on the fact that the line joining the midpoints of two sides of a triangle is parallel to the third side.
Let the three given points be A, B and C as in the figure, part . Connect them to form a triangle, as in part . Construct copies of the three lines of this triangle parallel to the sides but passing through the three vertices, as in . Finally extend the constructed lines so they meet, as in .
We now have a large triangle and four smaller triangles. All five triangles are similar, as one can easily prove using all those parallel lines. The four small triangles are congruent as they share sides in three pairs. That means that the midpoints of the large triangle are the three given points, making that the triangle we seek.