From Ceva to Kiepert and Jacobi
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.
Topic
Triangle centers
8 classical
The Euler Line from the Medial Triangle
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.
Four Orthocenters of a Cyclic Quadrilateral
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.
The Nine-Point Circle from the Circumcircle
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.
Tangents, Angle Bisectors, and a Common Point
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.
The Centroid through Ceva and Vectors
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.
The Incenter through Trigonometric Ceva
Angle-bisector concurrence and isogonal conjugation
Trigonometric Ceva proves that the three internal angle bisectors concur and shows why isogonal reflection preserves concurrence.
Two Views of the Orthocenter
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.