X1–X19

Classical & Ancient Centers

The foundational centers of classical triangle geometry: the incenter, centroid, circumcenter, orthocenter, nine-point center, symmedian point, Gergonne, Nagel, Mittenpunkt, Spieker, Feuerbach, Fermat, isodynamic, Napoleon, and Clawson points.
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c ( X_1 T_c ): Circle ( X_1 T_c ) ( X_3 A ): Circle ( X_3 A ) ( X_5 M_c ): Circle ( X_5 M_c ) [ X_2 X_3 ]: Euler Line [ X_1 X_2 ]: Nagel Line [ X_3 X_6 ]: Brocard Axis [ X_1 X_5 ]: Feuerbach Line [ X_2 X_7 ]: Centroid–Gergonne Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) T_c = Incircle Touchpoint T_c on AB (0.5714, 0.0000) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_7 = GERGONNE POINT (0.5903, 0.2769) X_8 = NAGEL POINT (0.4694, 0.1333) X_9 = MITTENPUNKT (0.5110, 0.1615) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_17 = 1st NAPOLEON POINT (0.5666, 0.2383) X_18 = 2nd NAPOLEON POINT (0.4434, 0.1317) X_19 = CLAWSON POINT (0.6246, 0.4244) X_4 = ORTHOCENTER (0.6122, 0.3958) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_2 = CENTROID (0.5374, 0.2000) X_1 = INCENTER (0.5714, 0.2333) A B C Ma Mb Mc Tc X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X17 X18 X19 X4 X3 X2 X1

X1–X19 (Classical & Ancient Centers) — Complete 19-center constellation constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

See also

Companion Topic Guides: Collection 1 Topics: Conjugacies, Central Lines, Circles & Triangles (\(T_1\)–\(T_{24}\))

X_1 — INCENTER

X1INCENTER

X1 is the point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, given by \(r = 2 \operatorname{area}(ABC)/(a + b + c)\).
X_2 — CENTROID

X2CENTROID

X2 is the point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X2 to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X2 and also balances atop any knife-edge that passes through X2. The triangles BXC, CXA, AXB have equal areas if and only if X = X2.
X_3 — CIRCUMCENTER

X3CIRCUMCENTER

X3 is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X3. This common distance is the radius of the circumcircle, which passes through vertices A, B, C. Called the circumradius, it is given by \(R = a/(2 \sin A) = abc/(4 \operatorname{area}(ABC))\).
X_4 — ORTHOCENTER

X4ORTHOCENTER

X4 is the point of concurrence of the altitudes of ABC.
X_5 — NINE-POINT CENTER

X5NINE-POINT CENTER

X5 is the center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X4, B-to-X4, C-to-X4. The radius of the nine-point circle is one-half the circumradius.
X_6 — SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)

X6SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)

X6 is the point of concurrence of the symmedians (i.e., reflections of medians in corresponding angle bisectors). X6 is the point which, when given by actual trilinear distances x,y,z, minimizes \(x^{2}\) + \(y^{2}\) + \(z^{2}\).
X_7 — GERGONNE POINT

X7GERGONNE POINT

Let Ta, Tb, Tc denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines ATa, BTb, CTc concur in X7.
X_8 — NAGEL POINT

X8NAGEL POINT

Let NaNbNc be the points in which the A-excircle meets BC, the B-excircle meets CA, and the C-excircle meets AB, respectively. The lines ANa, BNb, CNc concur in X8. Another construction of Na is to start at A and trace around ABC half its perimeter, and similarly for Nb and Nc. Also, X8 is the incenter of the anticomplementary triangle.
X_10 — SPIEKER CENTER

X10SPIEKER CENTER

The Spieker circle is the incircle of the medial triangle; its center, X10, is the centroid of the perimeter of ABC.
X_11 — FEUERBACH POINT

X11FEUERBACH POINT

X11 is the point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle. Feuerbach’s famous theorem states that the nine-point circle is tangent to the incircle and the three excircles.
X_13 — 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

X131st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

Construct the equilateral triangle BA'C having base BC and vertex A' on the negative side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA', BB', CC' concur in X13. If each of the angles A, B, C is < 2π/3, then X13 is the only center X for which the angles BXC, CXA, AXB are equal, and X13 minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X13 is equilateral.
X_14 — 2nd ISOGONIC CENTER

X142nd ISOGONIC CENTER

Construct the equilateral triangle BA'C having base BC and vertex A' on the positive side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X14. The antipedal triangle of X14 is equilateral.
X_15 — 1st ISODYNAMIC POINT

X151st ISODYNAMIC POINT

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. (This configuration, based on the cevians of X1, generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)
X_16 — 2nd ISODYNAMIC POINT

X162nd ISODYNAMIC POINT

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through a vertex and both isodynamic points. The pedal triangle of X16 is equilateral.
X_17 — 1st NAPOLEON POINT

X171st NAPOLEON POINT

Let Napa, Napb, Napc be the centers of the equilateral triangles in the construction of X13. The lines ANapa, BNapb, CNapc concur in X17.
X_18 — 2nd NAPOLEON POINT

X182nd NAPOLEON POINT

Let Napa, Napb, Napc be the centers of the equilateral triangles in the construction of X14. The lines ANapa, BNapb, CNapc concur in X18.
X_19 — CLAWSON POINT

X19CLAWSON POINT

X19 is the homothetic center of the orthic and extangents triangles. The Ayme triangle, constructed at X3610, is perspective to ABC with perspector X19.