In this presentation we develop a variety of terms related to the triangle and its circles and demonstrate these with applications of practical geometry using geometrical instruments. We introduce the concept of proof and show how geometrical proof can be used to validate the constructions. Similarly the constructions can be used to illustrate how proofs work. Students don’t get very much exposure to this form of practical mathematics. The development of geometrical software packages offers the opportunity to investigate geometry. However, hand constructions are invaluable aid to manual dexterity.

PRIMARY: perpendicular bisectors, angle bisectors, following geometrical instructions.

SECONDARY: plus points of concurrency of a triangle, Euler line, inscribed and circumscribed circles etc., simple theorems and proofs.

SIXTH FORM: plus Simpson line, pedal line, 9-point circle, further theorems and proofs.