Concepts

For a polyline $A\to E\to F\to B$ whose middle piece $EF$ is a "bridge" of fixed length and direction, minimize $AE + FB$ by translating $A$ along $\vec{EF}$ to $A'$; the minimum $|A'B|$ is achieved when $A', F, B$ are collinear.

Use axial symmetry to turn "sum of distances from a moving point on a line to two fixed points" into a single straight-segment problem between two fixed points.

In an isosceles figure with apex angle $2\alpha$, the "half-angle $\alpha$" drawn from the apex cuts off two polyline pieces; rotating one piece about the apex by $2\alpha$ aligns them into a single straight segment along the other side's extension.

Two isosceles triangles sharing a vertex; joining each pair of outer endpoints ("hands") produces a SAS-congruent pair of triangles, so the two "hand" segments are equal and the angle between them equals the shared apex angle.

When a moving point $P$ sees a fixed chord $AB$ at a fixed angle (90° included), the locus of $P$ is a fixed circle — draw the "invisible" circle, and distance extrema reduce to "moving point on a circle to a fixed point (or line)".

To minimize $PA + k \cdot PB$ ($0 < k < 1$), take an angle $\alpha = \arcsin k$, draw an auxiliary line through the fixed point, and convert the $k \cdot PB$ term into a perpendicular segment — the whole quantity collapses into "distance from a point to a line".

Two similar (not necessarily congruent) triangles sharing a vertex give a spiral similarity — a composite of rotation by $\alpha$ and dilation by $k$ centred at the shared vertex. The two "hand" segments satisfy $BD/AC = k$ with the angle between them equal to $\alpha$; the locus of a moving point retains its shape under this transformation ("plant melons, harvest melons").

Three equal angles on the same side of a single line force the two adjacent triangles into an AA-similar pair; use the corresponding-side ratios to solve for unknown lengths.

The composition of "rotation by $\theta$ about a fixed point" with "central dilation by $k$ from the same point"; a similarity transformation that simultaneously absorbs one rotation angle and one scaling factor — the core tool for "weighted-distance + angle" minimization.

To minimize $w_1 \cdot PA + w_2 \cdot PB + w_3 \cdot PC$, factor out $w_1$, then apply a spiral similarity at $C$ — rotate by $\theta$ and scale by $k = w_2/w_1$ — sending $B \to B''$. This converts $|PA| + k|PB| + w_3|PC|$ into the polyline $A \to P \to P'' \to B''$. The rotation angle $\theta$ is the angle opposite $w_3$ in the "weight triangle" $(w_1, w_2, w_3)$.

The locus of points whose distances to two fixed points have a constant ratio k≠1 is a circle; commonly used to construct an auxiliary point that converts "weighted-distance extrema" into plain-distance form.

The point inside a triangle that minimizes the sum of distances to its three vertices; when every interior angle is below 120°, the Fermat point sees all three vertices at 120° apart, and the minimum is computed via the 60° rotation construction.

For two fixed points $A, B$ on the same side of a line $\ell$ with a moving point $P\in\ell$, the maximum of $\angle APB$ is attained at the tangent point of the circle through $A, B$ that is tangent to $\ell$.

For any point $P$ in the plane, the "power" with respect to a fixed circle is an invariant; the product of the two segments cut by any chord (or tangent / secant) through $P$ equals the absolute value of $|PO|^2 - r^2$.

Between any two points in the plane, the straight segment is shortest; any polyline length is at least the straight-segment distance between its endpoints, with equality iff all points are collinear in order.

The branch of mathematics that studies spatial form and the properties of figures.