concurrency of angle bisectors, medians, perpendicular bisectors, altitudes When you think of a triangle the basic constructions you think of are perpendicular bisectors of the sides, angle bisectors, altitudes, and medians. Now if you were someone who just started learning about these concepts it would seem very unusual that all these basic constructions related to the triangle are concurrent. To me that seems really unlikely and I ask, is there some deep lying meaning or reason as to why this is so? Any help would be appreciated.

NOTE : I am not asking for the proofs of the concurrency theorems
The concept of concurrencies of various centers of a triangle can be appreciated from 3 different levels.

At the junior level, the said constructions are mainly done by the help of RULERs and PROTRACTORs. The point is to arouse interest and see if lines can be accurately or not.

At the intermediate level, lines should be drawn using STRAIGHT-EDGE and COMPASSES only. Techniques required are preliminary knowledge of congruency. Thus, the learning tasks are scheduled as studying congruency first and the topic of constructions right after it. The construction exercise can therefore also be considered as a consolidation of the topics on congruency.

At the advanced level, the truth of the concurrencies should be proved formally.
The concept of concurrencies of various centers of a triangle can be appreciated from 3 different levels.

At the junior level, the said constructions are mainly done by the help of RULERs and PROTRACTORs. The point is to arouse interest and see if lines can be accurately or not.

At the intermediate level, lines should be drawn using STRAIGHT-EDGE and COMPASSES only. Techniques required are preliminary knowledge of congruency. Thus, the learning tasks are scheduled as studying congruency first and the topic of constructions right after it. The construction exercise can therefore also be considered as a consolidation of the topics on congruency.

At the advanced level, the truth of the concurrencies should be proved formally.