Triangles and congruence

One of the most widespread objects in geometry are the triangles. The three congruence theorems (postulates) - SAS, ASA and SSS are the main tool in proving many famous and important theorems in the field of the Elementary Geometry and are very helpful for the better understanding of the isometries (rigid motions) of the plane - those transformations which preserve distance.

Problem 1. If rigid motion T leaves all the vertices of triangle ABCin place, then T is the identity transformation.

Solution. We must prove that every point D on the plane will stay fixed with respect to the transformation T. Suppose that the image of D is D'. Then it is clear that by SSS postulate we have $\Delta ABD \cong \Delta ABD'$. We can have either that $D\equiv D'$ or D is symmetric to D' with respect to AB. Suppose $D\neq D'$. Similarly we have that D is symmetric to D' with respect to AB, BC and AC, which contradicts to the assumption.

Problem 2. Point A is given inside a triangle. Draw a line segment with endpoints on the perimeter of the triangle so that the point divides the segment in half.

Solution. We can use a reflection with respect to the given point. After choosing a pair of intersection points of the triangle and its symmetric we will obtain the endpoints of the required segment. In general we will have three solutions.

Problem 3. Prove that if a triangle has two axes of symmetry, then it has at least three axes of symmetry.

Sketch of the Proof. If we reflect the first axes in the second easily follows that we will obtain a new axes of symmetry (except for the case when the axes are perpendicular). Suppose we have two perpendicular axes. Then all possible images of a vertex with respect to these axes form a rectangle, which is impossible.