Geometry

Inscribed angle theorem: inscribed angles subtended by the same arc are equal, and the central angle is twice the inscribed angle; the inscribed angle subtended by a diameter is $90^\circ$.
Cyclic quadrilateral: opposite angles are supplementary. Tangent–chord angle (angle between tangent and chord) equals the inscribed angle subtended by the intercepted arc.

Criteria for four points to be concyclic: for the same line segment, equal angles of subtension on the same side or supplementary angles of subtension on opposite sides $\iff$ the four points are concyclic — this is the core that links together 'feet of perpendiculars / right angles' (e.g. Simson line uses 'right angle ⇒ lies on the circle with the segment as diameter').

Similar triangles give the ratios of side lengths. Projection theorem: dropping an altitude onto the hypotenuse of a right-angled triangle gives $b^2=c\cdot p$, $a^2=c\cdot q$ (adding these gives Pythagoras).

Power of a point with respect to a circle: a secant/chord through a fixed point $P$ satisfies $PA\cdot PB=PC\cdot PD$; in the tangent case $PT^2=PA\cdot PB$. Ptolemy's theorem: for a cyclic quadrilateral $AC\cdot BD=AB\cdot CD+AD\cdot BC$ (proved by "taking a point on the diagonal with equal angles → two pairs of similar triangles → adding").

Two ways of computing the same area to establish an equation is the simplest and most effective method.

• Angle bisector theorem: $BD/DC=AB/AC$ (ratio of areas).
• Viviani's theorem: for a point inside an equilateral triangle, the sum of the distances to the three sides is $=$ the height (summing the areas of the three small triangles; since the three sides are equal in length they can be cancelled). For a general triangle it is $a\,d_a+b\,d_b+c\,d_c=2[ABC]$.

Place one vertex at the origin and one side on a coordinate axis, write the coordinates of the remaining points using side lengths and angles, then compute directly with the distance formula. The golden placement is "midpoint at the origin, opposite side on the $x$ axis", which greatly simplifies median/symmetry/perpendicular relationships and makes cross terms cancel automatically.

From this one can derive: the law of cosines $c^2=a^2+b^2-2ab\cos C$, Apollonius's median $m_a^2=\dfrac{2b^2+2c^2-a^2}{4}$. When synthetic methods give no idea for a while, the coordinate method can almost always push through to the end.

Taking the circumcentre $O$ as the origin ($|\mathbf a|=|\mathbf b|=|\mathbf c|=R$):

Thus many relations become one-line computations: $AH=2\,OM$ ($M$ is the midpoint of $BC$), the reflection of $H$ in $M$ is $=-\mathbf a$ (the antipode of $A$), the Euler line $O,G,H$ is collinear and $OG:GH=1:2$ (because $H=3G$), and the nine-point centre $N=\tfrac12(\mathbf a+\mathbf b+\mathbf c)$ is the midpoint of $OH$.

Equal tangent lengths: the incircle touches all three sides, and the two tangent lengths from each vertex are equal; let $AY=AZ=x,\ BZ=BX=y,\ CX=CY=z$, then $a=y+z$ etc., giving $x=s-a$ ($s$ is the semiperimeter).
Incentre angle: $\angle BIC=90^\circ+\tfrac12\angle A$ ($BI,CI$ bisects the angle + angle sum).

Simson line: $P$ on the circumcircle $\Rightarrow$ the feet of the perpendiculars from it to the three sides are collinear—use "right angle ⇒ concyclic on the circle with that diameter" to construct auxiliary circles, then track with directed angles.