Glossary¶
Canonical geometric definitions and concepts referenced across the 55 topics of the Fundamentals codex.
- Apollonian circles¶
- Apollonian circle¶
- circle of Apollonius¶
- circles of Apollonius¶
A family of circles defined as the locus of points whose distances from two fixed foci A and B have a constant ratio k, or the three circles of a triangle passing through a vertex and two isodynamic points.
- barycentric coordinates¶
- barycentric¶
- homogeneous barycentric coordinates¶
- barycentrics¶
Homogeneous coordinates (u : v : w) of a point P with respect to a reference triangle ABC, proportional to the signed areas of triangles PBC, PCA, and PAB.
- bilinear transformation¶
- bilinear transformations¶
- Möbius transformation¶
A fractional linear or projective mapping of the form w = (az + b)/(cz + d) that preserves cross-ratios and maps circles and lines to circles and lines.
- Carnot’s theorem¶
- Carnot theorem¶
A theorem relating the signed distances from the circumcenter O to the sides of triangle ABC with the inradius r and circumradius R: d_a + d_b + d_c = R + r.
- centers of similitude¶
- center of similitude¶
- homothetic center¶
- similitude center¶
The internal and external homothetic centers (\(S_+\) and \(S_-\)) of two circles, dividing the segment between their centers harmonically in the ratio of their radii.
- centroid¶
- medians¶
- median¶
The point of concurrency G = X(2) of the three medians of a triangle, dividing each median in a 2:1 ratio with barycentric coordinates 1 : 1 : 1.
- Ceva’s theorem¶
- cevian¶
- cevians¶
A criterion for three cevians AD, BE, CF of triangle ABC to concur: (BD/DC)(CE/EA)(AF/FB) = 1.
- circumcenter¶
- circumcircle¶
- circumradius¶
The center O = X(3) of the circumscribed circle of a triangle, located at the intersection of the perpendicular bisectors of the three sides.
- coaxial circles¶
- coaxial system¶
- pencil of circles¶
A one-parameter pencil of circles in which every pair shares the same radical axis; classified into intersecting (elliptic), non-intersecting (hyperbolic with limiting points), and tangent (parabolic) pencils.
- Conway triangle notation¶
- Conway notation¶
- Conway symbols¶
John H. Conway’s algebraic system for triangle geometry setting S = 2K (twice the triangle area) and S_A = (b^2 + c^2 - a^2)/2 = bc cos A, S_B = (c^2 + a^2 - b^2)/2, S_C = (a^2 + b^2 - c^2)/2.
- cyclic quadrilateral¶
- cyclic quad¶
A four-sided polygon inscribed in a single circle, satisfying opposite angle supplementary relations, Ptolemy’s theorem, and Brahmagupta’s area formula.
- ellipse¶
- elliptical¶
A conic section with eccentricity 0 <= e < 1 whose points have a constant sum of distances 2a from two fixed foci.
- equilateral triangle¶
- equilateral¶
A regular three-sided polygon with equal sides a = b = c and interior angles of 60 degrees, constructed in Euclid’s Proposition I.1.
- Euler line¶
The line passing through a non-equilateral triangle’s orthocenter H, nine-point center N, centroid G, and circumcenter O, with HG : GN : NO = 4 : 1 : 2.
- excircle¶
- excircles¶
- excenter¶
- excenters¶
- exradius¶
- extangent¶
One of three circles tangent to one side of a triangle and the extensions of the other two sides, centered at an excenter I_a, I_b, or I_c.
- Fermat point¶
- Torricelli point¶
- first Fermat point¶
The isogonic center F = X(13) minimizing the sum of distances PA + PB + PC to the vertices of a triangle with all angles < 120 degrees.
- geometric mean¶
- mean proportional¶
The mean proporional sqrt(ab) between two positive lengths a and b, constructed as the altitude to the hypotenuse of a right triangle with segments a and b.
- golden ratio¶
- divine proportion¶
- golden section¶
The extreme and mean ratio phi = (1 + sqrt(5))/2 ≈ 1.6180339887, satisfying phi^2 = phi + 1.
- golden triangle¶
- golden triangles¶
- golden gnomon¶
An isosceles triangle whose side-to-base ratio is the golden ratio phi, having vertex angles of 36-72-72 degrees (acute) or 108-36-36 degrees (gnomon).
- harmonic conjugate¶
- harmonic conjugates¶
- harmonic range¶
- harmonic division¶
Given collinear points A, B, C, the point D is the harmonic conjugate of C with respect to (A, B) when the cross-ratio (A, B; C, D) = -1, so AC/CB = AD/BD.
- harmonic mean¶
The reciprocal of the arithmetic mean of reciprocals, H = 2ab/(a + b), realized geometrically by the segment separating harmonic conjugates.
- Heron’s formula¶
- Heron formula¶
The expression for the area K of a triangle in terms of its semiperimeter s = (a+b+c)/2: K = sqrt(s(s-a)(s-b)(s-c)), or in biquadratic side form 16K^2 = 2a^2b^2 + 2b^2c^2 + 2c^2a^2 - a^4 - b^4 - c^4.
- incenter¶
- incircle¶
- inradius¶
The center I = X(1) of the inscribed circle of a triangle, located at the concurrency of the three interior angle bisectors with barycentric coordinates a : b : c.
- inversion¶
- inversive geometry¶
- circle inversion¶
A transformation with respect to a circle of center O and radius R mapping each point P to P’ on ray OP such that OP * OP’ = R^2.
- isosceles triangle¶
- isosceles¶
A triangle with at least two equal sides and two equal base angles (Euclid I.5, Pons Asinorum).
- isotomic conjugate¶
- isotomic¶
The point P’ obtained by reflecting the cevian traces of P across the midpoints of the corresponding sides, with barycentric coordinates 1/u : 1/v : 1/w.
- Menelaus’s theorem¶
- Menelaus theorem¶
- transversal¶
A collinearity criterion for three points D, E, F on the (extended) sides BC, CA, AB of triangle ABC: (BD/DC)(CE/EA)(AF/FB) = -1.
- nine-point circle¶
- Feuerbach circle¶
- nine-point center¶
Feuerbach’s circle centered at N = X(5) of radius R/2 passing through the three side midpoints, the three altitude feet, and the three Euler midpoints between the vertices and the orthocenter.
- orthic triangle¶
The pedal triangle H_A H_B H_C of the orthocenter H, formed by the feet of the three altitudes of triangle ABC; it minimizes perimeter among all inscribed triangles of an acute triangle (Fagnano’s problem).
- orthocenter¶
- altitude¶
- altitudes¶
The concurrency H = X(4) of the three altitudes of a triangle, with barycentric coordinates tan A : tan B : tan C = 1/S_A : 1/S_B : 1/S_C.
- orthogonal circles¶
- orthogonality¶
Two circles that intersect at right angles, so that the radii to their intersection points are tangent to the other circle and d^2 = r_1^2 + r_2^2.
- parabola¶
- parabolas¶
A conic section of eccentricity e = 1 defined as the locus of points equidistant from a focus F and a directrix line.
- pedal triangle¶
- pedal triangles¶
The triangle formed by the perpendicular projections of a point P onto the three sides of reference triangle ABC.
- pole and polar¶
- pole¶
- polar¶
- pole-polar¶
A projective duality with respect to a conic or circle mapping a point P (the pole) to the line of harmonic conjugates or chord of contact (the polar).
- power of a point¶
- chord intersection¶
For a point P and circle of center O and radius R, the invariant quantity Pi(P) = OP^2 - R^2 equal to the signed product PA * PB along any secant line through P.
- Pythagorean triple¶
- Pythagorean triples¶
A tuple of positive integers (a, b, c) satisfying a^2 + b^2 = c^2, parametrized by Euclid’s formula a = m^2 - n^2, b = 2mn, c = m^2 + n^2.
- radical axis¶
- radical axes¶
- radical center¶
The locus of points having equal power with respect to two non-concentric circles, forming a line perpendicular to the line of centers.
- Stewart’s theorem¶
- Stewart theorem¶
A theorem relating the length d of a cevian AD dividing side BC = a into segments m and n: b^2 m + c^2 n = a(d^2 + mn).
- symmedian point¶
- symmedian¶
- Lemoine point¶
The Lemoine point K = X(6), the concurrency of the reflections of the medians across the angle bisectors, with barycentric coordinates a^2 : b^2 : c^2.
- trilinear coordinates¶
- trilinear¶
- trilinears¶
Homogeneous coordinates (alpha : beta : gamma) proportional to the signed directed distances from a point P to the sidelines BC, CA, AB, related to barycentrics by (u : v : w) = (a*alpha : b*beta : c*gamma).
- vesica piscis¶
- vesica¶
The lens-shaped intersection of two congruent circles each passing through the other’s center, fundamental to Euclid I.1 and Gothic pointed arch tracery with height-to-width ratio sqrt(3).
Kimberling Triangle Centers (ETC Index)¶
X(1) — Incenter¶
Concurrency of the three interior angle bisectors; barycentrics a : b : c. (GEOMETOR ETC X(1)).
X(2) — Centroid¶
Concurrency of the three medians; barycentrics 1 : 1 : 1. (GEOMETOR ETC X(2)).
X(3) — Circumcenter¶
Concurrency of the three perpendicular bisectors; barycentrics a^2 S_A : b^2 S_B : c^2 S_C. (GEOMETOR ETC X(3)).
X(4) — Orthocenter¶
Concurrency of the three altitudes; barycentrics S_B S_C : S_C S_A : S_A S_B. (GEOMETOR ETC X(4)).
X(5) — Nine-Point Center¶
Midpoint of O and H on the Euler line; center of Feuerbach’s nine-point circle. (GEOMETOR ETC X(5)).
X(6) — Symmedian (Lemoine) Point¶
Isogonal conjugate of the centroid G; barycentrics a^2 : b^2 : c^2. (GEOMETOR ETC X(6)).
X(7) — Gergonne Point¶
Concurrency of cevians to the incircle contact points T_A, T_B, T_C. (GEOMETOR ETC X(7)).
X(8) — Nagel Point¶
Concurrency of cevians to the excircle touchpoints; isotomic conjugate of X(7). (GEOMETOR ETC X(8)).
X(9) — Mittenpunkt¶
Symmedian point of the excentral triangle; perspector of the excentral and medial triangles. (GEOMETOR ETC X(9)).
X(11) — Feuerbach Point¶
Point of internal tangency between the incircle and the nine-point circle. (GEOMETOR ETC X(11)).
X(13) — First Fermat (Torricelli) Point¶
First isogonic center minimizing \(|PA| + |PB| + |PC|\) for angles < 120 degrees. (GEOMETOR ETC X(13)).
X(14) — Second Fermat Point¶
Second isogonic center constructed via inwardly directed equilateral triangles. (GEOMETOR ETC X(14)).
X(15) — First Isodynamic Point¶
Isogonal conjugate of X(13); common intersection of the three Apollonian circles. (GEOMETOR ETC X(15)).
X(16) — Second Isodynamic Point¶
Isogonal conjugate of X(14); second common intersection of the three Apollonian circles. (GEOMETOR ETC X(16)).
X(20) — De Longchamps Point¶
Reflection of the orthocenter H across the circumcenter O on the Euler line. (GEOMETOR ETC X(20)).
X(26) — Circumcenter of the Tangential Triangle¶
Center of the circle circumscribing the tangential triangle. (GEOMETOR ETC X(26)).
X(40) — Bevan Point¶
Circumcenter of the excentral triangle I_A I_B I_C; reflection of I across O. (GEOMETOR ETC X(40)).
X(55) — Internal Center of Similitude of Incircle and Circumcircle¶
Insimilicenter dividing segment OI internally in ratio R : r. (GEOMETOR ETC X(55)).
X(56) — External Center of Similitude of Incircle and Circumcircle¶
Exsimilicenter dividing segment OI externally in ratio R : -r. (GEOMETOR ETC X(56)).