Those who have the privilege of contributing new proofs of old theorems to the columns of the Gazette do so, normally, with complete confidence in the validity of the new proofs but possibly with some doubts as to whether the new proofs will be regarded as better than the old ones. The following attempt to set out new proofs for certain properties of the triangle differs from the norm in that there is no doubt whatever that the new proofs will be regarded as worse than the old ones and the only doubt is whether they will be regarded as valid. The doubt may arise because the aim is to deduce the truth of certain general theorems from the proved truth of one or two particular cases of the theorems. This is far from being the usual procedure in mathematics, though it is not without precedent; for example, if we show that ax 2 + bx + c is zero for three values of x , it follows that it is zero for any and all values of x .