Given a plane X, a point M in the plane and a point N not in the plane.
Find all points R in X such that the ratio (NM + MR)/NR is the maximum.
(In reply to
solution by Daniel)
My solution seems to have a different isosceles triangle: one where MN = MR, and the locus is only one point, the third vertex of this triangle lying on a plane containing M and N, perpendicular to plane X.
If the locus were a circle on plane X, one would expect two maxima to be achieved on the intersection of the above-mentioned plane with plane X, but I get a table with only the maximum I found:
dist.from
M ratio MN MR NR
-7.00000 -0.09451 5.83095 -7.00000 12.36932
-6.99000 -0.09378 5.83095 -6.99000 12.35962
-6.98000 -0.09304 5.83095 -6.98000 12.34991
-6.97000 -0.09230 5.83095 -6.97000 12.34021
-6.96000 -0.09157 5.83095 -6.96000 12.33051
-6.95000 -0.09083 5.83095 -6.95000 12.32082
-6.94000 -0.09009 5.83095 -6.94000 12.31112
-6.93000 -0.08934 5.83095 -6.93000 12.30142
-6.92000 -0.08860 5.83095 -6.92000 12.29172
-6.91000 -0.08786 5.83095 -6.91000 12.28202
-6.90000 -0.08711 5.83095 -6.90000 12.27233
-6.89000 -0.08636 5.83095 -6.89000 12.26263
-6.88000 -0.08562 5.83095 -6.88000 12.25293
-6.87000 -0.08487 5.83095 -6.87000 12.24324
-6.86000 -0.08412 5.83095 -6.86000 12.23354
-6.85000 -0.08337 5.83095 -6.85000 12.22385
-6.84000 -0.08261 5.83095 -6.84000 12.21416
-6.83000 -0.08186 5.83095 -6.83000 12.20446
-6.82000 -0.08110 5.83095 -6.82000 12.19477
-6.81000 -0.08035 5.83095 -6.81000 12.18508
-6.80000 -0.07959 5.83095 -6.80000 12.17539
-6.79000 -0.07883 5.83095 -6.79000 12.16569
-6.78000 -0.07807 5.83095 -6.78000 12.15600
-6.77000 -0.07731 5.83095 -6.77000 12.14631
-6.76000 -0.07655 5.83095 -6.76000 12.13662
-6.75000 -0.07579 5.83095 -6.75000 12.12693
-6.74000 -0.07502 5.83095 -6.74000 12.11724
-6.73000 -0.07426 5.83095 -6.73000 12.10756
-6.72000 -0.07349 5.83095 -6.72000 12.09787
-6.71000 -0.07272 5.83095 -6.71000 12.08818
-6.70000 -0.07195 5.83095 -6.70000 12.07849
-6.69000 -0.07118 5.83095 -6.69000 12.06881
-6.68000 -0.07041 5.83095 -6.68000 12.05912
-6.67000 -0.06963 5.83095 -6.67000 12.04944
-6.66000 -0.06886 5.83095 -6.66000 12.03975
-6.65000 -0.06808 5.83095 -6.65000 12.03007
-6.64000 -0.06731 5.83095 -6.64000 12.02038
-6.63000 -0.06653 5.83095 -6.63000 12.01070
-6.62000 -0.06575 5.83095 -6.62000 12.00102
-6.61000 -0.06497 5.83095 -6.61000 11.99133
-6.60000 -0.06419 5.83095 -6.60000 11.98165
-6.59000 -0.06340 5.83095 -6.59000 11.97197
-6.58000 -0.06262 5.83095 -6.58000 11.96229
-6.57000 -0.06183 5.83095 -6.57000 11.95261
-6.56000 -0.06104 5.83095 -6.56000 11.94293
-6.55000 -0.06026 5.83095 -6.55000 11.93325
-6.54000 -0.05947 5.83095 -6.54000 11.92357
-6.53000 -0.05868 5.83095 -6.53000 11.91390
-6.52000 -0.05788 5.83095 -6.52000 11.90422
-6.51000 -0.05709 5.83095 -6.51000 11.89454
-6.50000 -0.05629 5.83095 -6.50000 11.88486
-6.49000 -0.05550 5.83095 -6.49000 11.87519
-6.48000 -0.05470 5.83095 -6.48000 11.86551
-6.47000 -0.05390 5.83095 -6.47000 11.85584
-6.46000 -0.05310 5.83095 -6.46000 11.84616
-6.45000 -0.05230 5.83095 -6.45000 11.83649
-6.44000 -0.05150 5.83095 -6.44000 11.82682
-6.43000 -0.05069 5.83095 -6.43000 11.81714
-6.42000 -0.04989 5.83095 -6.42000 11.80747
-6.41000 -0.04908 5.83095 -6.41000 11.79780
-6.40000 -0.04827 5.83095 -6.40000 11.78813
-6.39000 -0.04746 5.83095 -6.39000 11.77846
-6.38000 -0.04665 5.83095 -6.38000 11.76879
-6.37000 -0.04584 5.83095 -6.37000 11.75912
-6.36000 -0.04503 5.83095 -6.36000 11.74945
-6.35000 -0.04421 5.83095 -6.35000 11.73978
-6.34000 -0.04340 5.83095 -6.34000 11.73012
-6.33000 -0.04258 5.83095 -6.33000 11.72045
-6.32000 -0.04176 5.83095 -6.32000 11.71078
-6.31000 -0.04094 5.83095 -6.31000 11.70112
-6.30000 -0.04012 5.83095 -6.30000 11.69145
-6.29000 -0.03930 5.83095 -6.29000 11.68178
-6.28000 -0.03847 5.83095 -6.28000 11.67212
-6.27000 -0.03765 5.83095 -6.27000 11.66246
-6.26000 -0.03682 5.83095 -6.26000 11.65279
-6.25000 -0.03599 5.83095 -6.25000 11.64313
-6.24000 -0.03516 5.83095 -6.24000 11.63347
-6.23000 -0.03433 5.83095 -6.23000 11.62381
-6.22000 -0.03350 5.83095 -6.22000 11.61415
-6.21000 -0.03266 5.83095 -6.21000 11.60449
-6.20000 -0.03183 5.83095 -6.20000 11.59483
-6.19000 -0.03099 5.83095 -6.19000 11.58517
-6.18000 -0.03015 5.83095 -6.18000 11.57551
-6.17000 -0.02931 5.83095 -6.17000 11.56585
-6.16000 -0.02847 5.83095 -6.16000 11.55619
-6.15000 -0.02763 5.83095 -6.15000 11.54654
-6.14000 -0.02679 5.83095 -6.14000 11.53688
-6.13000 -0.02594 5.83095 -6.13000 11.52722
-6.12000 -0.02510 5.83095 -6.12000 11.51757
-6.11000 -0.02425 5.83095 -6.11000 11.50791
-6.10000 -0.02340 5.83095 -6.10000 11.49826
-6.09000 -0.02255 5.83095 -6.09000 11.48861
-6.08000 -0.02170 5.83095 -6.08000 11.47895
-6.07000 -0.02084 5.83095 -6.07000 11.46930
-6.06000 -0.01999 5.83095 -6.06000 11.45965
-6.05000 -0.01913 5.83095 -6.05000 11.45000
-6.04000 -0.01827 5.83095 -6.04000 11.44035
-6.03000 -0.01741 5.83095 -6.03000 11.43070
-6.02000 -0.01655 5.83095 -6.02000 11.42105
-6.01000 -0.01569 5.83095 -6.01000 11.41140
-6.00000 -0.01483 5.83095 -6.00000 11.40175
-5.99000 -0.01396 5.83095 -5.99000 11.39211
-5.98000 -0.01309 5.83095 -5.98000 11.38246
-5.97000 -0.01223 5.83095 -5.97000 11.37281
-5.96000 -0.01136 5.83095 -5.96000 11.36317
-5.95000 -0.01049 5.83095 -5.95000 11.35352
-5.94000 -0.00961 5.83095 -5.94000 11.34388
-5.93000 -0.00874 5.83095 -5.93000 11.33424
-5.92000 -0.00786 5.83095 -5.92000 11.32459
-5.91000 -0.00699 5.83095 -5.91000 11.31495
-5.90000 -0.00611 5.83095 -5.90000 11.30531
-5.89000 -0.00523 5.83095 -5.89000 11.29567
-5.88000 -0.00435 5.83095 -5.88000 11.28603
-5.87000 -0.00346 5.83095 -5.87000 11.27639
-5.86000 -0.00258 5.83095 -5.86000 11.26675
-5.85000 -0.00169 5.83095 -5.85000 11.25711
-5.84000 -0.00080 5.83095 -5.84000 11.24747
-5.83000 0.00008 5.83095 -5.83000 11.23783
-5.82000 0.00098 5.83095 -5.82000 11.22820
-5.81000 0.00187 5.83095 -5.81000 11.21856
-5.80000 0.00276 5.83095 -5.80000 11.20893
-5.79000 0.00366 5.83095 -5.79000 11.19929
-5.78000 0.00455 5.83095 -5.78000 11.18966
-5.77000 0.00545 5.83095 -5.77000 11.18002
-5.76000 0.00635 5.83095 -5.76000 11.17039
-5.75000 0.00725 5.83095 -5.75000 11.16076
-5.74000 0.00816 5.83095 -5.74000 11.15113
-5.73000 0.00906 5.83095 -5.73000 11.14149
-5.72000 0.00997 5.83095 -5.72000 11.13186
-5.71000 0.01087 5.83095 -5.71000 11.12223
-5.70000 0.01178 5.83095 -5.70000 11.11261
-5.69000 0.01269 5.83095 -5.69000 11.10298
-5.68000 0.01361 5.83095 -5.68000 11.09335
-5.67000 0.01452 5.83095 -5.67000 11.08372
-5.66000 0.01544 5.83095 -5.66000 11.07410
-5.65000 0.01635 5.83095 -5.65000 11.06447
-5.64000 0.01727 5.83095 -5.64000 11.05485
-5.63000 0.01819 5.83095 -5.63000 11.04522
-5.62000 0.01912 5.83095 -5.62000 11.03560
-5.61000 0.02004 5.83095 -5.61000 11.02597
-5.60000 0.02096 5.83095 -5.60000 11.01635
-5.59000 0.02189 5.83095 -5.59000 11.00673
-5.58000 0.02282 5.83095 -5.58000 10.99711
-5.57000 0.02375 5.83095 -5.57000 10.98749
-5.56000 0.02468 5.83095 -5.56000 10.97787
-5.55000 0.02562 5.83095 -5.55000 10.96825
-5.54000 0.02655 5.83095 -5.54000 10.95863
-5.53000 0.02749 5.83095 -5.53000 10.94901
-5.52000 0.02842 5.83095 -5.52000 10.93940
-5.51000 0.02936 5.83095 -5.51000 10.92978
-5.50000 0.03031 5.83095 -5.50000 10.92016
-5.49000 0.03125 5.83095 -5.49000 10.91055
-5.48000 0.03219 5.83095 -5.48000 10.90094
-5.47000 0.03314 5.83095 -5.47000 10.89132
-5.46000 0.03409 5.83095 -5.46000 10.88171
-5.45000 0.03504 5.83095 -5.45000 10.87210
-5.44000 0.03599 5.83095 -5.44000 10.86249
-5.43000 0.03694 5.83095 -5.43000 10.85288
-5.42000 0.03790 5.83095 -5.42000 10.84327
-5.41000 0.03886 5.83095 -5.41000 10.83366
-5.40000 0.03981 5.83095 -5.40000 10.82405
-5.39000 0.04077 5.83095 -5.39000 10.81444
-5.38000 0.04174 5.83095 -5.38000 10.80483
-5.37000 0.04270 5.83095 -5.37000 10.79523
-5.36000 0.04366 5.83095 -5.36000 10.78562
-5.35000 0.04463 5.83095 -5.35000 10.77602
-5.34000 0.04560 5.83095 -5.34000 10.76641
-5.33000 0.04657 5.83095 -5.33000 10.75681
-5.32000 0.04754 5.83095 -5.32000 10.74720
-5.31000 0.04852 5.83095 -5.31000 10.73760
-5.30000 0.04949 5.83095 -5.30000 10.72800
-5.29000 0.05047 5.83095 -5.29000 10.71840
-5.28000 0.05145 5.83095 -5.28000 10.70880
-5.27000 0.05243 5.83095 -5.27000 10.69920
-5.26000 0.05341 5.83095 -5.26000 10.68960
-5.25000 0.05440 5.83095 -5.25000 10.68000
-5.24000 0.05538 5.83095 -5.24000 10.67041
-5.23000 0.05637 5.83095 -5.23000 10.66081
-5.22000 0.05736 5.83095 -5.22000 10.65122
-5.21000 0.05835 5.83095 -5.21000 10.64162
-5.20000 0.05934 5.83095 -5.20000 10.63203
-5.19000 0.06034 5.83095 -5.19000 10.62243
-5.18000 0.06134 5.83095 -5.18000 10.61284
-5.17000 0.06233 5.83095 -5.17000 10.60325
-5.16000 0.06334 5.83095 -5.16000 10.59366
-5.15000 0.06434 5.83095 -5.15000 10.58407
-5.14000 0.06534 5.83095 -5.14000 10.57448
-5.13000 0.06635 5.83095 -5.13000 10.56489
-5.12000 0.06735 5.83095 -5.12000 10.55530
-5.11000 0.06836 5.83095 -5.11000 10.54571
-5.10000 0.06938 5.83095 -5.10000 10.53613
-5.09000 0.07039 5.83095 -5.09000 10.52654
-5.08000 0.07140 5.83095 -5.08000 10.51696
-5.07000 0.07242 5.83095 -5.07000 10.50737
-5.06000 0.07344 5.83095 -5.06000 10.49779
-5.05000 0.07446 5.83095 -5.05000 10.48821
-5.04000 0.07548 5.83095 -5.04000 10.47863
-5.03000 0.07651 5.83095 -5.03000 10.46904
-5.02000 0.07753 5.83095 -5.02000 10.45946
-5.01000 0.07856 5.83095 -5.01000 10.44989
-5.00000 0.07959 5.83095 -5.00000 10.44031
-4.99000 0.08062 5.83095 -4.99000 10.43073
-4.98000 0.08166 5.83095 -4.98000 10.42115
-4.97000 0.08269 5.83095 -4.97000 10.41158
-4.96000 0.08373 5.83095 -4.96000 10.40200
-4.95000 0.08477 5.83095 -4.95000 10.39243
-4.94000 0.08581 5.83095 -4.94000 10.38285
-4.93000 0.08685 5.83095 -4.93000 10.37328
-4.92000 0.08790 5.83095 -4.92000 10.36371
-4.91000 0.08895 5.83095 -4.91000 10.35413
-4.90000 0.08999 5.83095 -4.90000 10.34456
-4.89000 0.09105 5.83095 -4.89000 10.33499
-4.88000 0.09210 5.83095 -4.88000 10.32542
-4.87000 0.09315 5.83095 -4.87000 10.31586
-4.86000 0.09421 5.83095 -4.86000 10.30629
-4.85000 0.09527 5.83095 -4.85000 10.29672
-4.84000 0.09633 5.83095 -4.84000 10.28716
-4.83000 0.09739 5.83095 -4.83000 10.27759
-4.82000 0.09846 5.83095 -4.82000 10.26803
-4.81000 0.09952 5.83095 -4.81000 10.25846
-4.80000 0.10059 5.83095 -4.80000 10.24890
-4.79000 0.10166 5.83095 -4.79000 10.23934
-4.78000 0.10273 5.83095 -4.78000 10.22978
-4.77000 0.10381 5.83095 -4.77000 10.22022
-4.76000 0.10489 5.83095 -4.76000 10.21066
-4.75000 0.10596 5.83095 -4.75000 10.20110
-4.74000 0.10704 5.83095 -4.74000 10.19155
-4.73000 0.10813 5.83095 -4.73000 10.18199
-4.72000 0.10921 5.83095 -4.72000 10.17243
-4.71000 0.11030 5.83095 -4.71000 10.16288
-4.70000 0.11139 5.83095 -4.70000 10.15332
-4.69000 0.11248 5.83095 -4.69000 10.14377
-4.68000 0.11357 5.83095 -4.68000 10.13422
-4.67000 0.11467 5.83095 -4.67000 10.12467
-4.66000 0.11576 5.83095 -4.66000 10.11512
-4.65000 0.11686 5.83095 -4.65000 10.10557
-4.64000 0.11796 5.83095 -4.64000 10.09602
-4.63000 0.11907 5.83095 -4.63000 10.08647
-4.62000 0.12017 5.83095 -4.62000 10.07692
-4.61000 0.12128 5.83095 -4.61000 10.06738
-4.60000 0.12239 5.83095 -4.60000 10.05783
-4.59000 0.12350 5.83095 -4.59000 10.04829
-4.58000 0.12461 5.83095 -4.58000 10.03874
-4.57000 0.12573 5.83095 -4.57000 10.02920
-4.56000 0.12685 5.83095 -4.56000 10.01966
-4.55000 0.12797 5.83095 -4.55000 10.01012
-4.54000 0.12909 5.83095 -4.54000 10.00058
-4.53000 0.13021 5.83095 -4.53000 9.99104
-4.52000 0.13134 5.83095 -4.52000 9.98150
-4.51000 0.13247 5.83095 -4.51000 9.97197
-4.50000 0.13360 5.83095 -4.50000 9.96243
-4.49000 0.13473 5.83095 -4.49000 9.95289
-4.48000 0.13586 5.83095 -4.48000 9.94336
-4.47000 0.13700 5.83095 -4.47000 9.93383
-4.46000 0.13814 5.83095 -4.46000 9.92429
-4.45000 0.13928 5.83095 -4.45000 9.91476
-4.44000 0.14043 5.83095 -4.44000 9.90523
-4.43000 0.14157 5.83095 -4.43000 9.89570
-4.42000 0.14272 5.83095 -4.42000 9.88617
-4.41000 0.14387 5.83095 -4.41000 9.87664
-4.40000 0.14502 5.83095 -4.40000 9.86712
-4.39000 0.14618 5.83095 -4.39000 9.85759
-4.38000 0.14733 5.83095 -4.38000 9.84807
-4.37000 0.14849 5.83095 -4.37000 9.83854
-4.36000 0.14965 5.83095 -4.36000 9.82902
-4.35000 0.15082 5.83095 -4.35000 9.81950
-4.34000 0.15198 5.83095 -4.34000 9.80997
-4.33000 0.15315 5.83095 -4.33000 9.80045
-4.32000 0.15432 5.83095 -4.32000 9.79093
-4.31000 0.15549 5.83095 -4.31000 9.78142
-4.30000 0.15667 5.83095 -4.30000 9.77190
-4.29000 0.15785 5.83095 -4.29000 9.76238
-4.28000 0.15903 5.83095 -4.28000 9.75287
-4.27000 0.16021 5.83095 -4.27000 9.74335
-4.26000 0.16139 5.83095 -4.26000 9.73384
-4.25000 0.16258 5.83095 -4.25000 9.72433
-4.24000 0.16377 5.83095 -4.24000 9.71481
-4.23000 0.16496 5.83095 -4.23000 9.70530
-4.22000 0.16615 5.83095 -4.22000 9.69579
-4.21000 0.16735 5.83095 -4.21000 9.68628
-4.20000 0.16854 5.83095 -4.20000 9.67678
-4.19000 0.16974 5.83095 -4.19000 9.66727
-4.18000 0.17095 5.83095 -4.18000 9.65776
-4.17000 0.17215 5.83095 -4.17000 9.64826
-4.16000 0.17336 5.83095 -4.16000 9.63876
-4.15000 0.17457 5.83095 -4.15000 9.62925
-4.14000 0.17578 5.83095 -4.14000 9.61975
-4.13000 0.17699 5.83095 -4.13000 9.61025
-4.12000 0.17821 5.83095 -4.12000 9.60075
-4.11000 0.17943 5.83095 -4.11000 9.59125
-4.10000 0.18065 5.83095 -4.10000 9.58175
-4.09000 0.18187 5.83095 -4.09000 9.57226
-4.08000 0.18310 5.83095 -4.08000 9.56276
-4.07000 0.18433 5.83095 -4.07000 9.55327
-4.06000 0.18556 5.83095 -4.06000 9.54377
-4.05000 0.18679 5.83095 -4.05000 9.53428
-4.04000 0.18803 5.83095 -4.04000 9.52479
-4.03000 0.18927 5.83095 -4.03000 9.51530
-4.02000 0.19051 5.83095 -4.02000 9.50581
-4.01000 0.19175 5.83095 -4.01000 9.49632
-4.00000 0.19300 5.83095 -4.00000 9.48683
-3.99000 0.19425 5.83095 -3.99000 9.47735
-3.98000 0.19550 5.83095 -3.98000 9.46786
-3.97000 0.19675 5.83095 -3.97000 9.45838
-3.96000 0.19801 5.83095 -3.96000 9.44889
-3.95000 0.19927 5.83095 -3.95000 9.43941
-3.94000 0.20053 5.83095 -3.94000 9.42993
-3.93000 0.20179 5.83095 -3.93000 9.42045
-3.92000 0.20306 5.83095 -3.92000 9.41097
-3.91000 0.20432 5.83095 -3.91000 9.40149
-3.90000 0.20559 5.83095 -3.90000 9.39202
-3.89000 0.20687 5.83095 -3.89000 9.38254
-3.88000 0.20814 5.83095 -3.88000 9.37307
-3.87000 0.20942 5.83095 -3.87000 9.36359
-3.86000 0.21070 5.83095 -3.86000 9.35412
-3.85000 0.21199 5.83095 -3.85000 9.34465
-3.84000 0.21327 5.83095 -3.84000 9.33518
-3.83000 0.21456 5.83095 -3.83000 9.32571
-3.82000 0.21585 5.83095 -3.82000 9.31624
-3.81000 0.21715 5.83095 -3.81000 9.30678
-3.80000 0.21845 5.83095 -3.80000 9.29731
-3.79000 0.21974 5.83095 -3.79000 9.28785
-3.78000 0.22105 5.83095 -3.78000 9.27838
-3.77000 0.22235 5.83095 -3.77000 9.26892
-3.76000 0.22366 5.83095 -3.76000 9.25946
-3.75000 0.22497 5.83095 -3.75000 9.25000
-3.74000 0.22628 5.83095 -3.74000 9.24054
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Posted by Charlie
on 2015-06-05 15:59:04 |
re: solution
