let ΔABC be an equilateral triangle. let P be an interior point (strictly interior, not on any border) of ΔABC. let ∠PAB=a1, ∠PBC=a2, ∠PCA=a3, ∠PAC=b1, ∠PBA=b2, ∠PCB=b3. i am interested in finding all possible cases such that:
- a1, b1, a2, b2, a3, b3 are all distinct
- a1/b1, a2/b2, a3/b3 are all rational
my motivation for former condition is to exclude the angle bisector(s) which yield trivial (so uninteresting) cases.
one set of solution which is not so difficult to find is (a1, a2, a3)=(π/30, 7π/30, 4π/15). the corresponding (b1, b2, b3)=(3π/10, π/10, π/15). we can express this in a simpler form: (a1, b1, a2, b2, a3, b3)=(1, 9, 7, 3, 8, 2)π/30
the common denominator “30” can be interpreted as the “resolution”. we divide the straight angle into 30 equal portions. 1+9+7+3+8+2=30
with the help of computer programming i found 4 more sets of solution:
- (537, 19583, 11074, 9046, 19477, 643)π/60360
- (6908, 26764, 18678, 14994, 25519, 8153)π/101016
- (324, 38090, 34838, 3576, 35809, 2605)π/115242
- (4086, 39380, 29250, 14216, 36037, 7429)π/130398
it’s quite unexpected that the next set of solution after resolution=30 occurs at resolution=60360
my conjecture: there are infinitely many sets of solution. my question: is there any analytical method to generate all sets of solution, instead of solely relying on brute force of computer programming?