ABC-triangles

If we talk about the centre of a triangle, what might we be referring to? Any triangle has many different points that could regarded as its centre; in fact, Encyclopedia of Triangle Centres lists over 70 000 possibilities. Three of the most famous centres, that every triangle will possess (although they may coincide), are the incentre (where the three angle bisectors meet), the centroid (where the three medians meet) and the orthocentre (where the three altitudes meet). Proofs that these centres are well-defined and exist for every triangle are simple and satisfying, good examples of reasoning (if we are teachers) for our students. Proving the three altitudes of a triangle share a point using the scalar product of vectors is a wonderful demonstration of the power of this idea.