Following up various suggestions for scalene test triangles T, I computed the conic determinant (normalised for scale invariance) when the "centre" S is the centroid and the 6 subsidiary points are incentres: the result quantifies relatively how far the hexad departs from "conicity" (as it were) for T with vertices shown. 0 {[0, 0], [0, 1], [1, 0]} right isosceles 1.6e-8 {[3, 4], [-12, 5], [9, -9]} WFL old test 9.7e-8 {[0, 0], [0, 12], [15, 20]} WFL sides {12, 17, 25} -1.4e-7 {[0, 0], [0, 1], [3^(1/2), 0]} DA angles {1/6, 1/3, 1/2} -6.6e-8 {[0, 0], [0, 1], [0.256001685, 0.124419097]} KM sides ~{263, 697, 924} Precision is 50 decimals; notice how small all the discrepancies are. Convincing win for Dan's remarkably simple suggestion then --- which I had also considered, but discarded as a result of somehow mistakenly concluding that the conjecture succeeds for right-angled cases, instead of isosceles as explained earlier by PJCM . [WRM's ingenious suggestion --- minimising maximal overlap with the mirror-image triangle --- must await more determined analysis.] Incidentally, the rational {12, 17, 25}-sided triangle turns out to have a rational incentre at [10/3, 10] --- I haven't looked at other special points. Fred Lunnon