Classical
Classical Euclidean configurations, proofs, and the structures revealed by different methods.
Two proofs and the further reach of each method
Ceva’s theorem proves the concurrence of the medians and leads naturally to isotomic conjugation. Vectors locate the centroid and reveal why another triangle can share it.
Angle-bisector concurrence and isogonal conjugation
Trigonometric Ceva proves that the three internal angle bisectors concur and shows why isogonal reflection preserves concurrence.
Cyclic quadrilaterals, vectors, and the Euler line
Two cyclic quadrilaterals explain why the third altitude is forced through the intersection of the first two, while a circumcenter-based vector formula reveals the Euler line.
A homothety centered at the centroid
The circumcenter of a triangle is the orthocenter of its medial triangle, and a centroid-centered homothety carries the circumcenter to the original orthocenter.
A hidden half-turn
For a cyclic quadrilateral, the four orthocenters obtained by omitting one vertex at a time are the images of the original vertices under a single half-turn.
Two short proofs from quadratic equations
Subtracting circle equations produces the radical axes; adding normalized equations of two parabolas with perpendicular axes produces a circle through their four intersections.
Three formulas revealing one homothety
In the unit-circle model, three formulas for the side midpoints, altitude feet, and vertex–orthocenter midpoints reveal a single homothety carrying the circumcircle to the nine-point circle.
Two regular side constructions encoded by fixed rotations
Complex multiplication encodes the external equilateral triangles of Napoleon’s theorem and the external squares of Van Aubel’s theorem, reducing both conclusions to rotation identities.
Side ratios, angle ratios, and external concurrence
Trigonometric Ceva, derived by the sine rule, proves Kiepert’s, Jacobi’s, and Kariya’s concurrence theorems, with the Gergonne point as Kariya’s boundary case.
A barycentric route to X(55)
A barycentric calculation locates the common point of three lines joining tangent intersections to angle-bisector traces, then identifies it as the isogonal conjugate of the Gergonne point.
Perpendicular feet and Carnot’s equal-angle extension
Two cyclic quadrilaterals align the perpendicular feet of a point on the circumcircle. The same angle argument gives Carnot’s extension to oblique projections.