X3614
1st HUTSON-FEUERBACH POINT
Let
Eua Eub Euc be the
Feuerbach triangle , L the line through
A and
X5 , and
A″ = L∩
Eub Euc ; define
B″ and
C″ cyclically. Then the lines
Eua A″ ,
Eub B″ ,
Euc C″ concur in
X3614 .
X3614 = {
X5 ,
X12 }-harmonic conjugate of
X11 . Further,
X3614 is the
trilinear pole (with respect to the Feuerbach triangle) of the perspectrix of
ABC and the Feuerbach triangle. (
Randy Hutson , August 30, 2011.)
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 5, 12 }: Feuerbach Line { 1, 5 }: Feuerbach Line { 35, 546 }: Central Line L(35, 546) { 381, 498 }: Central Line L(381, 498) { 442, 1155 }: Central Line L(442, 1155)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_1 = INCENTER (0.5714, 0.2333) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_546 = MIDPOINT OF X(4) AND X(5) (0.5842, 0.3223) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5518, 0.2140) X_442 = COMPLEMENT OF SCHIFFLER POINT (0.5418, 0.2114) X_1155 = SCHRÖDER POINT (1.7143, 2.3328) X_3614 = 1st HUTSON-FEUERBACH POINT (0.5602, 0.2447)
A
B
C
X11
X5
X12
X1
X35
X546
X381
X498
X442
X1155
X3614
(a, b, c) = (5, 6, 7) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 5, 12 }: Feuerbach Line { 1, 5 }: Feuerbach Line { 35, 546 }: Central Line L(35, 546) { 381, 498 }: Central Line L(381, 498) { 442, 1155 }: Central Line L(442, 1155)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5867, 0.2178) X_1 = INCENTER (0.6000, 0.2000) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5556, 0.1111) X_546 = MIDPOINT OF X(4) AND X(5) (0.6050, 0.3600) X_381 = MIDPOINT OF X(2) AND X(4) (0.5933, 0.3200) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5709, 0.1782) X_442 = COMPLEMENT OF SCHIFFLER POINT (0.5516, 0.1768) X_1155 = SCHRÖDER POINT (1.2000, 1.4000) X_3614 = 1st HUTSON-FEUERBACH POINT (0.5788, 0.2282)
A
B
C
X11
X5
X12
X1
X35
X546
X381
X498
X442
X1155
X3614
(a, b, c) = (3, 4, 5) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 5, 12 }: Feuerbach Line { 1, 5 }: Feuerbach Line { 35, 546 }: Central Line L(35, 546) { 381, 498 }: Central Line L(381, 498) { 442, 1155 }: Central Line L(442, 1155)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5729, 0.2203) X_1 = INCENTER (0.5841, 0.2021) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5465, 0.1118) X_546 = MIDPOINT OF X(4) AND X(5) (0.5885, 0.3644) X_381 = MIDPOINT OF X(2) AND X(4) (0.5787, 0.3239) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5596, 0.1801) X_442 = COMPLEMENT OF SCHIFFLER POINT (0.5435, 0.1791) X_1155 = SCHRÖDER POINT (1.1159, 1.4809) X_3614 = 1st HUTSON-FEUERBACH POINT (0.5663, 0.2310)
A
B
C
X11
X5
X12
X1
X35
X546
X381
X498
X442
X1155
X3614
Kepler (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 5, 12 }: Feuerbach Line { 1, 5 }: Feuerbach Line { 35, 546 }: Central Line L(35, 546) { 381, 498 }: Central Line L(381, 498) { 442, 1155 }: Central Line L(442, 1155)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.6001, 0.1762) X_1 = INCENTER (0.6154, 0.1300) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5793, 0.0033) X_546 = MIDPOINT OF X(4) AND X(5) (0.5999, 0.5534) X_381 = MIDPOINT OF X(2) AND X(4) (0.5888, 0.4614) X_498 = YFF CONCURRENT CONGRUENT CIRCLES POINT (0.5865, 0.1151) X_442 = COMPLEMENT OF SCHIFFLER POINT (0.5473, 0.1176) X_1155 = SCHRÖDER POINT (0.7604, 0.6385) X_3614 = 1st HUTSON-FEUERBACH POINT (0.5872, 0.2151)
A
B
C
X11
X5
X12
X1
X35
X546
X381
X498
X442
X1155
X3614
Kimberling (6, 9, 13) | AB = 1
X3614 (1st HUTSON-FEUERBACH POINT ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(1 + 3 \cos(B - C) : 1 + 3 \cos(C - A) : 1 + 3 \cos(A - B)\)
Barycentrics: \((\sin A)(1 + 3 \cos(B - C)) : (\sin B)(1 + 3 \cos(C - A)) : (\sin C)(1 + 3 \cos(A - B))\)
Search Value (6, 9, 13): \(1.033305322267\)
▸ Alternate Coordinates1 form
▸ Linear Combinations1 combination
▸ Incident Central Lines14 lines
{1 5 35 546 55 3091 56 3090 140 3585 381 498 442 1155 899 3136 1329 2476 1478 1656 1698 1836 2475 3035 2635 3142 3058 3085
Notes
See also X3615 5954 .
Let Eua Eub Euc be the Feuerbach triangle. Let A″ be the trilinear pole, wrt Eua Eub Euc , of line BC ; define B″ and C″ cyclically. The lines Eua A″ , Eub B″ , Euc C″ concur in X3614
