Content of type 'topic'

F.1 Albert’s Parabola

Given a parabola y = x^2 and a point P = (P_x, P_y) on it, the tangent line T and normal line N at P satisfy y = 2 P_x x - P_x^2 and y = -x/(2 P_x) + P_x^2 + 1/2; requiring N to pass through (M, 0) yields the depressed cubic P_x^3 + (1/2) P_x - M/2 = 0.

F.2 Bilinear Transformations

The canonical bilinear transformation B(x) = (1 + x)/(1 - x) and its inverse B^{-1}(y) = (y - 1)/(y + 1) link the internal division ratio kappa = x/r of a segment AB to the Apollonian distance ratio k = (1 + kappa)/(1 - kappa).

F.3 Biquadratic Forms in Heron’s Formula

Under the linear substitution X = a/2, W = (b + c)/2, and Y = (c - b)/2, Heron’s area formula reduces to the biquadratic product K = sqrt((W^2 - X^2)(X^2 - Y^2)).

F.4 Carnot’s Theorem

In any triangle ABC, the sum of the signed perpendicular distances from the circumcenter O to the three sides BC, CA, AB equals the sum of the circumradius R and the inradius r: d_a + d_b + d_c = R + r.

F.5 Centers of Similitude

Given two circles (O_1, r_1) and (O_2, r_2) separated by center distance d = |O_1 O_2| with r_2 > r_1, their common internal and external tangents concur at the internal and external centers of similitude S_IN and S_EX, which divide O_1 O_2 harmonically in the ratio r_1 : r_2.

F.6 Centroid

The three medians AM_A, BM_B, CM_C joining the vertices of triangle ABC to the midpoints of the opposite sides concur at the centroid G = X(2), which divides each median in a 2 : 1 ratio and has homogeneous barycentric coordinates 1 : 1 : 1.

F.7 Ceva and Menelaus

Three cevians AD, BE, CF of triangle ABC concur at a point P if and only if (AF/FB)(BD/DC)(CE/EA) = +1 (Ceva’s Theorem), whereas points D, E, F’ on the sidelines are collinear if and only if (AF’/F’B)(BD/DC)(CE/EA) = -1 (Menelaus’s Theorem), making (A, B; F, F’) a harmonic range.

F.8 Chord Intersection

If two chords AB and CD of a circle (O, R) intersect at a point X at distance r = |OX| from the center, then the product of their segments is constant and equals the power of X with respect to the circle: |AX|*|BX| = |CX|*|DX| = R^2 - |OX|^2.

F.9 Circle Equations

Pairwise subtraction of Cartesian circle equations (x - u_i)^2 + (y - v_i)^2 = r_i^2 eliminates the quadratic terms x^2 + y^2 to yield linear radical-axis relations, determining both the side Z of a square from three corner distances and the radius of a Sangaku circle tangent to an inscribed square and equilateral triangle.

F.10 Circles and Geometric Mean

In a semicircle on diameter AB = m + n, the perpendicular altitude h = CH from a point C on the circumference to AB is the geometric mean sqrt(m*n) of the segments AH = m and HB = n, while the radius OC = (m + n)/2 is their arithmetic mean.

F.11 Circumcenter

The perpendicular bisectors of the three sides of triangle ABC concur at the circumcenter O = X(3), equidistant from A, B, C by the circumradius R = abc/(4K), with homogeneous barycentric coordinates a^2 S_A : b^2 S_B : c^2 S_C.

F.12 Coaxial Circles

A hyperbolic pencil of coaxial circles with limiting points L_1 = (-r, 0) and L_2 = (+r, 0) coincides with the Apollonian equipotential circles d_1 / d_2 = k around two point charges Q_1, Q_2, sharing the y-axis as their common radical axis.

F.13 Compare Triangles in Circles

For a family of triangles ABC inscribed in a fixed circle of radius R sharing a fixed chord AB = c (with central angle 2*phi where sin(phi) = c/(2R)), the area K decomposes into the signed sum of three central isosceles triangles: K = (1/2) c h_c + (1/2) a h_a - (1/2) b h_b.

F.14 Conway Notation

In John H. Conway’s triangle notation, setting S = 2K (twice the area of triangle ABC), S_omega = (a^2 + b^2 + c^2)/2, and S_A = (b^2 + c^2 - a^2)/2, S_B = (c^2 + a^2 - b^2)/2, S_C = (a^2 + b^2 - c^2)/2 reduces trigonometric and barycentric identities to polynomial expressions satisfying S_B S_C + S_C S_A + S_A S_B = S^2.

F.15 Cyclic Quadrilateral

A cyclic quadrilateral WXYZ whose interior angles form the arithmetic progression theta, 2*theta, 3*theta, 4*theta has theta = 36 degrees (W = 36, X = 72, Y = 144, Z = 108) and golden-ratio diagonals |XZ| = 2 sin(36) = sqrt(3 - phi) and |WY| = 2 sin(72) = phi*sqrt(3 - phi) in the unit circle; by contrast, no non-trivial cyclic quadrilateral can have interior angles in geometric progression theta, k*theta, k^2*theta, k^3*theta.

F.16 d’Alembert’s Theorem

Given three circles of distinct radii in the plane, their six pairwise internal and external centers of similitude lie by threes on four projective lines L_{+++}, L_{–+}, L_{-+-}, and L_{+–} (the Monge–d’Alembert similitude axes).

F.17 Ellipse

Given semi-major axis a and semi-minor axis b, a four-center circular-arc oval approximating the ellipse x^2/a^2 + y^2/b^2 = 1 is constructed by a 7-step five-circle compass procedure, while the exact conic satisfies F = a*e, d = a/e, and vertex mean proportional V = a = sqrt(d*F).

F.18 Equilateral Triangle

In an equilateral triangle of side a, the incenter, centroid, circumcenter, and orthocenter coincide, with altitude h_A = (sqrt(3)/2)a = 3r and R = 2r; moreover, points A’, B’, C’ placed along the angle bisectors AI, BI, CI of any triangle ABC at distances A’I = (2r/sqrt(3)) sin(alpha/2 + pi/6) form an equilateral triangle.

F.19 Equal Incircles Theorem

If a fan of rays from apex P to a baseline cuts triangle ABC into consecutive subtriangles whose incircles have equal radii r_1 = r_2 = … = r_n = r, then the ray angles gamma_n with the altitude PR (of height h) follow the hyperbolic progression tan(gamma_n) = sinh(a + n*b) with tanh(b/2) = r/(h - r), so overlapping pairs of subtriangles also have equal incircles.

F.20 Extangent

The external common tangents to the incircle and circumcircle (or excircles) bound the extangents triangle, while Euler’s relation d_{IO}^2 = R^2 - 2rR has derivative d(d_{IO})/dR = (R - r)/sqrt(R^2 - 2rR).

F.21 Fermat Point

For any triangle ABC whose largest interior angle is less than 120 degrees, erecting external equilateral triangles BCX_A, CAY_B, ABZ_C on its sides yields concurrent segments AX_A, BY_B, CZ_C that intersect at the Fermat-Torricelli point F = X(13), where <AFB = <BFC = <CFA = 120 degrees and FA + FB + FC is minimized.

F.22 Functions

The standard normal cumulative distribution Phi(x) and error function erf(x) satisfy Phi(x) = 1/2 + (1/2) erf(x/sqrt(2)), where erf(x) is approximated to 1.5e-7 accuracy by Abramowitz and Stegun’s 5-term rational-exponential formula.

F.23 Golden Triangles

In the acute golden isosceles triangle ABC with base AB = 1, legs AC = BC = phi, and angles 72-72-36 degrees, bisecting base angle B meets AC at D such that CD = 1, DA = phi - 1 = 1/phi, and BD = AB = 1, partitioning ABC into a smaller golden triangle BCD and a golden gnomon ABD.

F.24 Gothic Arch

In a square ABCD of side s with pointed Gothic arch arcs of radius s centered at A(0, 0) and B(s, 0) meeting at apex E(s/2, (sqrt(3)/2)s), an inscribed tracery circle centered at P(s/2, Q) of radius r = s - p satisfies p*s = P_x H_x + P_y H_y at its tangency point H.

F.25 Harmonic Conjugates

Collinear points C and D divide segment AB (of midpoint M and half-length r) internally and externally in the same ratio k = |AC|/|BC| = |AD|/|BD| if and only if |MC|*|MD| = r^2, which occurs if and only if the circle on diameter AB and the Apollonian circle on diameter CD intersect orthogonally.

F.26 Harmonic Conjugate Inversion

For a circle Gamma of center M and radius r on diameter AB, internal point C at x = |MC| and its harmonic conjugate D at y = |MD| are inverses satisfying x*y = r^2, while the tangent DT = T forms a parallelogram M D M_T T whose long diagonal z satisfies z^2 = y^2 + 3*r^2 by Stewart’s Theorem.

F.27 Harmonic Mean

If C and D are harmonic conjugates with respect to segment AB (so |AC|/|CB| = |AD|/|BD|), then the length |CD| is the harmonic mean of |BD| and |AD|: |CD| = 2 |BD| |AD| / (|BD| + |AD|) = r (gamma + 1/gamma).

F.28 Incenter

The three interior angle bisectors of triangle ABC concur at the incenter I = X(1) with homogeneous barycentric coordinates a : b : c, inradius r = K/s satisfying 1/r = 1/h_a + 1/h_b + 1/h_c, and Euler’s distance d_{IO}^2 = R(R - 2r).

F.29 Inradius

The inradius r = K / s = (1/2) sqrt(((b+c-a)(c+a-b)(a+b-c))/(a+b+c)) = 4 R sin(alpha/2) sin(beta/2) sin(gamma/2) takes integer values on primitive Pythagorean triples and partitions the perimeter into Conway segments x, y, z whose touchpoints T_A, T_B, T_C concur at the Gergonne point Ge = X(7).

F.30 Isosceles Triangles

An isosceles triangle of given semiperimeter s and area K has half-base x = c/2 governed by the cubic 2 s x^3 - s^2 x^2 + K^2 = 0, which under x = y + s/6 reduces to Cardano’s depressed cubic y^3 - (s^2/12) y - (s^3/54 - K^2/(2s)) = 0.

F.31 Logic Tests

Given collinear points A, B, C, the three Boolean inequality tests T_1 = (|CA| < |AB|), T_2 = (|CB| < |AB|), and T_3 = (|CA| < |CB|) uniquely partition the line AB into Region A (-1 < k < 0), interior segment AB (0 < k < infty), and Region B (-infty < k < -1), determining the sign of k = +/- |AC| / |BC|.

F.32 Markowski’s Theorem

If J, K, L denote the perpendicular sagittas from the side midpoints of triangle ABC to the circumcircle, Markowski’s Theorem states J K L = (1/2) r^2 R = (R^2 - d_IO^2)^2 / (8 R).

F.33 Euclidean (m, n) Generators

For integers m > n >= 1, Euclid’s quadratic generators a = m^2 - n^2, b = 2 m n, and c = m^2 + n^2 satisfy a^2 + b^2 = c^2 and inradius r = n(m - n).

F.34 Nine-Point Circle

The circle Gamma_N passing through the three altitude feet H_A, H_B, H_C also passes through the three side midpoints M_A, M_B, M_C and the three Euler midpoints; its radius is r_N = R/2 and its center N = X(5) bisects segment OH on the Euler line.

F.35 Obtuse and Acute Triangles

By Euclid’s Elements II.12 (excess) and II.13 (deficit), the square on side a opposite an obtuse or acute angle satisfies a^2 = b^2 + c^2 +/- 2 c x, where x = |a^2 - (b^2 + c^2)| / (2 c) is the projection of side b onto line c and h = sqrt(b^2 - x^2) is the altitude.

F.36 Orthic Triangle

The orthic triangle H_A H_B H_C formed by the altitude feet of an acute triangle ABC is the pedal triangle of the orthocenter H = X(4), is homothetic to the tangential triangle, has H as its incenter, and solves Fagnano’s 1775 problem by minimizing the perimeter among all triangles inscribed in ABC.

F.37 Euler’s Inradius Parabola

For a fixed circumradius R, Euler’s relation d^2 = R^2 - 2 r R defines a downward-opening parabola r(d) = R/2 - d^2/(2R) = (R - d)(R + d)/(2R) with vertex (0, R/2) at the equilateral triangle and roots at the degenerate limits d = +/- R.

F.38 Pedal Triangles

The pedal triangle A’B’C’ of a point P(tau_A : tau_B : tau_C) has area Delta(P) = ((R^2 - OP^2)/(4 R^2)) Delta, and its six side-projection segments satisfy Carnot’s alternating square identity AC’^2 + BA’^2 + CB’^2 = C’B^2 + A’C^2 + B’A^2.

F.39 Pole and Polar with Respect to a Triangle

Given a triangle ABC and a point P with cevian traces A’, B’, C’, the intersections X = BC cap B’C’, Y = CA cap C’A’, and Z = AB cap A’B’ are collinear on the trilinear polar L of the pole P.

F.40 Regular Polygons and Nested Radicals

For a regular N-gon inscribed in a unit circle (R = 1), doubling the number of sides maps edge length E_{N/2} to E_N = sqrt(2 - sqrt(4 - E_{N/2}^2)), generating closed-form nested quadratic radicals for the 2^k, 3*2^k, 5*2^k, and 15*2^k polygon families.

F.41 Geometric Problems and Extremal Loci

Among all triangles with longest side c = |AB| = 1, the acute locus inside the vesica piscis above the semicircle on AB has area A = int_0^{sqrt(3)} (sqrt(4 - x^2) - 1) dx - pi/4 = (5 pi - 6 sqrt(3))/12 approx 0.44297 (for c = 2), and maximizing d_IO over acute triangles approaches the degenerate right-isosceles limit C -> (0, 1).

F.42 Altitude of a Pythagorean Triple Triangle

In a right triangle with legs a, b and hypotenuse c, the altitude h_c = a b / c satisfies the Inverse Pythagorean Theorem 1/h_c^2 = 1/a^2 + 1/b^2 and divides the hypotenuse into projections u = b^2/c and y = a^2/c with h_c^2 = u y.

F.43 Radical Axes and Radical Center

The radical axis of two circles of radii r_1, r_2 with center separation d cuts the line of centers at x = d/2 + (r_1^2 - r_2^2)/(2 d) from O_1, and the three radical axes of the vertex-altitude circles Gamma_A(h_A), Gamma_B(h_B), Gamma_C(h_C) concur at the radical center W.

F.44 Homothetic Ratios and Crossed Diagonals

Given two parallel uprights of heights h_1 < h_2 separated by horizontal distance d and viewed from their external center of similitude O, the crossed diagonals meet at horizontal distance x = d h_1 / (h_1 + h_2) from the first upright, while O lies at distance y - x = d h_1 / (h_2 - h_1) to its left, forming a harmonic range.

F.45 Squares, Subdivided Lattices, and the Japanese Theorem

A square inscribed in a circle of radius R has side s = sqrt(2) R; inside a square of side s, intersecting cevians divide the interior in exact rational proportions, and by Carnot’s Theorem the sum of inradii in any triangulation of a cyclic polygon is invariant.

F.46 Stewart’s Theorem

If a cevian of length f from C divides base c = AB into segments k’ c and k c (with k + k’ = 1), Stewart’s Theorem states f^2 = k a^2 + k’ b^2 - k k’ c^2; at the midpoint k = k’ = 1/2, this yields Apollonius’s median formula f^2 = (1/2)(a^2 + b^2 - c^2/2).

F.47 The UVW Coordinate Triangle

Placing the altitude foot H_C at the origin (0, 0) with A = (-U, 0), B = (W, 0), and C = (0, V) parameterizes any triangle by three numbers (U, V, W) with a = sqrt(V^2 + W^2), b = sqrt(U^2 + V^2), c = U + W, and reduces the cevian traces D, E, F through P(p, q) to explicit 2x2 rational matrix formulas.

F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety

If lines parallel to the sides of triangle ABC are drawn tangent to the incircle, the three corner subtriangles are homothetic to ABC with scale factors x/s, y/s, z/s (where x = s - a, y = s - b, z = s - c), so their inradii satisfy r_A + r_B + r_C = r.

F.49 Vesica Piscis and Acute-Obtuse Probability

When segment AB of length c is the longest side of triangle ABC, vertex C lies inside the vesica piscis of area c^2 (4 pi - 3 sqrt(3))/6; the semicircle of diameter AB partitions each quadrant of area c^2 (4 pi - 3 sqrt(3))/24 into an obtuse region (O) = c^2 pi / 16 and an acute region (A) = c^2 (5 pi - 6 sqrt(3))/48, giving an exact obtuse-to-acute probability ratio (O)/(A) = 3 pi / (5 pi - 6 sqrt(3)) approx 1.773.

F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem

Given the three altitudes h_A, h_B, h_C of triangle ABC with reciprocal semiperimeter H = (h_A^{-1} + h_B^{-1} + h_C^{-1})/2, Mitchell’s reciprocal Heron formula states K^{-1} = 4 sqrt(H (H - h_A^{-1})(H - h_B^{-1})(H - h_C^{-1})), and a triangle is right-angled at C if and only if h_A^{-2} + h_B^{-2} = h_C^{-2} (the Symphonic Theorem).

F.51 Apollonian Circles and the Problem of Apollonius

The locus of points P with distance ratio |PA| / |PB| = k > 1 (where |AB| = d) is the Apollonian circle of radius R = d k / (k^2 - 1) centered at distance M’ = d / (k^2 - 1) beyond B, cutting line AB at the internal and external harmonic conjugates C and D such that |A T’|^2 = |AC| |AD| = |AB| |A M’|.

F.52 Homogeneous Barycentric Coordinates

Every triangle center X(n) has homogeneous barycentric coordinates (u : v : w) normalizable to absolute weights lambda_A + lambda_B + lambda_C = 1 such that X = lambda_A A + lambda_B B + lambda_C C; for the nine-point center N = X(5), lambda_A = 1/4 + (b^2 - c^2)(a^2 - (b^2 + c^2)) / (32 K^2).

F.53 Geometric Half-Angle Trigonometry

In a unit semicircle of radius R = 1 and diameter AB = 2, a point P at central angle theta has altitude sin(theta), base projections 1 + cos(theta) and 1 - cos(theta), and chord |AP| = 2 cos(theta/2) = sqrt(2 + 2 cos(theta)), giving cos^2(theta/2) = (1 + cos(theta))/2 and sin^2(theta/2) = (1 - cos(theta))/2.

F.54 HP-35s Triangle Program and Coordinate Scripts

In the canonical frame A = (0, 0), B = (M, 0), C = (I, J), the forward and inverse barycentric-Cartesian transformations BARY 2 CAR and CAR 2 BARY are given by P = M U + I V, Q = J V and V = Q / J, U = (P - I V) / M, T = 1 - (U + V).

F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction

Given the three interior angles (alpha, beta, gamma) of a triangle normalized to unit area K = 1 (so s = 1/r), the Conway intouch segments x = s - a = r cot(alpha/2), y = s - b = r cot(beta/2), and z = s - c = r cot(gamma/2) determine the side lengths a = y + z, b = z + x, and c = x + y = 2 s - (a + b).