A geometry diagram is not merely an illustration of the givens. It is a workspace for making hidden structure visible.

Draw relationships, not decoration

An auxiliary line earns its place when it creates a reusable relationship:

• a pair of equal angles;
• two similar triangles;
• a cyclic quadrilateral;
• a midpoint or parallel line;
• a reflection, rotation, or homothety.

Before adding a line, name the relationship you hope it will create. This keeps the diagram from becoming a dense collection of guesses.

Work backward from the desired fact

If the goal is an angle equality, ask which cyclic quadrilateral would make it immediate. If the goal is a length equality, ask whether an isosceles triangle, reflection, or circle radius could produce it.

The useful question is often not “what can I prove from this picture?” but “what picture would make the desired statement routine?”

Keep one clean copy

Maintain a sparse base diagram and add one experimental construction at a time. A crowded diagram hides the very incidences and symmetries it was meant to reveal.

This is also why the diagrams in this library stay visually restrained: emphasis should encode mathematical relevance, not decoration.