Proving a theorem is not a matter of talent: it is a matter of method. Here we present a systematic approach, inspired by the classical Euclidean tradition (Aref, Problems in plane geometry).
In brief — A scheme for geometry proofs
- Draw an accurate figure. A good figure suggests the ideas; a bad figure hides them. Put the labels () on the main points.
- Write down Hypotheses and Thesis. Separate sharply what is given (Hp) from what you must prove (Th). It is the most important step.
- Think backwards (Backward Thinking). Ask yourself: “To prove the Thesis, what would it suffice for me to know?” and work back towards the Hypotheses.
- Look for auxiliary constructions. Often you need to add a segment, extend a side, draw a parallel or an altitude. The common constructions are:
- extending a side;
- drawing a parallel to a side through a vertex;
- drawing an altitude, a median or an angle bisector;
- joining two points to form new triangles.
- Identify the triangles. Almost all plane geometry proofs reduce to showing that two triangles are congruent or similar. Look for two triangles with elements in common.
- Justify every step. Every statement must be motivated: “by hypothesis”, “by construction”, “by the first criterion”, “because vertically opposite”, etc.
Links
Topics: Euclidean geometry
Concepts: Congruence criteria · Proof · Hypothesis and thesis
Skills: Proving · Synthetic geometry