The purpose of this study is to examine the decision-making process of eliminating triangle construction and to discuss why it is necessary to teach triangle construction in elementary mathematics. The construction of triangles in elementary school was introduced in the 4th curriculum and continued until the 2007 revised curriculum in Korea. Triangle construction was removed from the 2009 revised curriculum, and completely removed from textbooks published under the 2015 revised curriculum. In this paper, I claim that triangle construction is necessary for discovering the conditions determining triangles, and it is a very useful mathematical material for developing ability to manage rulers, protractors, and compasses, and developing mathematical thinking.