Construction of triangles by various points

Given three points that are not all on a straight line, there is exactly one triangle with those points as its vertices. Sometimes, you are given other sets of three points, such as the midpoints of the sides, the feet of the altitudes, or the points where the angle bisectors meet the sides. The task is to construct the triangle that has these points as its special points.

Important properties

• If the three given points are the vertices, simply connect them.

• If the three given points are midpoints of the sides, you can reconstruct the triangle using the midpoint theorem and parallel lines.

• If the three given points are feet of altitudes (the orthocenter's pedal triangle), you can reconstruct the original triangle using properties of altitudes.

• Not all sets of three points determine a triangle uniquely; the points must satisfy certain geometric conditions.