Well-spoken by Miles

Puzzle explanation

The letters omitted from wordplay spelt “Euler line”. The three occurrences of X in the grid were the vertices of a tall scalene triangle. The preamble phrase LONE TEPEE LEANS (a jumble of the unclued entries ÉLAN, STEP, LEO, EEN) hinted at this triangle’s shape. The nine occurrences of O marked the defining points of the triangle’s nine-point circle: the mid-points of the sides, the feet of the altitudes (where a line from a vertex meets the opposite side at a right angle), and the mid-points along the altitudes between the vertices and the orthocentre (where the three altitudes meet). In the centre of the circle (the shape both delineated by Os and resembling an O), a fourth X was added to mark the “nine-point centre”, at the cell corner between the L of LIANA and the P of IN PERSON. The Euler line had to be drawn parallel to the triangle’s right-hand side and passing through this X (beside the spaced out letters of EULER, not through their cell centres). The puzzle’s title cryptically referred to Euler as a homophone of “oiler”, an oil well.

Clue explanations

Conventions: * = anagram, < = reversal

BEAUXITE BÊTE around (A + UZI with X for Z)
ETEN ET + EN; the de Longchamps point is also on the Euler line
NASUTE SAN < + UTE
BENIGHTM(E)NT (MIGHTN’T BE) * around N
UNEQUALS ALS after (UNE + QU)
KEFIR (E(gg) F(ermented)) in KIR
TENACE TEN + ACE (flanking J, Q, K)
SO(U)PER SOPER; ref. Donald Soper
DECOL(L)ATORS DECORATORS with L for R
CATNEP (PET CAN) *
APOGEE APE(x) around (O + GE)
LIANA (A + NAIL) <
IN PERSON IN + PER + SO + N
NEAR-SIGHTED (IN + NEAR-SIGHTED) * = DISHEARTENING
(E)SLOIN NOILS *
SOLE 2 meanings; E after SOL
TANDOORS ROAST * around (ND + O’)
(R)EBUKED BE < + UK + ED
CTENE C + TENE
BENEFACTOR (ROT + CAFÉ + NEB) <
AUGURA(L) (ARUGULA − L) <
UNHATS HAUNTS with first two letters moved to the middle
(I)SMS SMS
ETNA 2 meanings
LETHE LIFE − IF + THE
CAPOTASTOS CAPO + TAST(e) + O’ + S
NUTS NU + T(ank)S
CERE SINCERE − SIN
LEAP 2 meanings
NONG SONG with N for S
ALLEGRO (liber)AL LEGRO(om)
SPANDEX SPAN + homophone of DECKS
PEPINO PEP + (ON + I) <
AN(N)EX NE in AX
NOEL(E) “lover” with NO EL = “over”
BEST 3 meanings
IRON I + RN around O