$$\begin{gathered}D,\ E \text{ midpoints of } AB,\ AC\\[2pt] \Rightarrow\ DE \parallel BC \ \text{ and } \ DE = \tfrac{1}{2}BC\end{gathered}$$

Converse holds too: midpoint + line parallel to the third side lands on the other midpoint. Reflex: Spot two midpoints in a triangle? Instantly know two things about that joining line — no calculation needed.

$$\angle ARP = \angle PQR \quad (\text{alt. seg thm})$$

Tangent meets chord → that angle equals the angle in the alternate segment (far arc). Reflex: Spot tangent+chord, then jump your eye to the far arc for the match.

Congruency — same size

Tests: SSS, ASA, SAS, AAS, RHS.

Beware the fake test — ASS (two sides + a non-included angle) proves nothing.

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Similarity — same shape

Tests: AA, SSS (ratios), SAS (ratios + included angle).

Reflex: Quote the test by name — equal sides prove congruent, sides in ratio prove similar.

• Angle at centre $= 2 \times$ angle at circumference
• Angle in a semicircle $= 90^\circ$
• Angles in the same segment are equal
• Opposite angles of a cyclic quadrilateral add to $180^\circ$
• Tangent $\perp$ radius at the point of contact
• Tangents from an external point are equal
• Perpendicular from the centre to a chord bisects the chord
• Equal chords are equidistant from the centre
• Tangent–chord angle $=$ angle in the alternate segment

Reflex: Every proof step needs a named reason — spot which fact fits, then quote it by name.

Read the goal first, then let each clue in the figure pick its tool:

• Parallel lines → alternate, corresponding, interior angles; look for similar triangles (AA).
• Diameter → angle in a semicircle $= 90^\circ$.
• Tangent → radius $\perp$ tangent; tangent–chord $=$ alternate segment; two tangents → isosceles triangle.
• Cyclic quadrilateral → opposite angles add to $180^\circ$.
• Two midpoints → midpoint theorem: parallel and half.
• Equal sides or radii → isosceles base angles.
• Prove similar → hunt two equal angles (AA); need a length → similar-triangle ratios.
• Stuck → label one angle $a$ and mark every angle equal to it round the figure.

Reflex: Every statement needs a reason — no reason, no marks.