Geometry · Triangle construction · Midline · Thales circle · Altitude · Case discussion
Two points \(A_1\), \(B_1\) and a line \(p\) are given in the plane. Construct a triangle \(ABC\) in which \(A_1\) is the midpoint of the side \(BC\), \(B_1\) is the midpoint of the side \(CA\), and the altitude from the vertex \(A\) lies on the line \(p\).
Describe the construction, prove that it produces such a triangle, and discuss for which positions of the data a solution exists and how many there are.
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Serbian Regional Competition (Okruzno takmicenje) 2003, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source