In “Trans. R.S.E.” (1864–5) Fox Talbot proved very simply, by means of a species of co-ordinates depending on confocal conics, the following theorem, at the same time asking for a simple geometrical proof.If two sets of three concentric circles, with the same common difference of radii, intersect one another—the chords of the arcs intercepted on the mean circle of each series by the extremes of the other are equal.
The usual proofs of Desargues Theorem employ either metrical or analytical methods of projection from a point outside the plane; and if it is attempted to translate the analytical proof by the von Stuadt-Reye methods, the result is very long and there is trouble with coincidences. It is the object of this note to give a short geometrical proof which, in addition to the usual axioms of incidence and extension, uses only the assumption that a projectivity which leaves three points on a line unchanged also leaves all points on it unchanged. Degenerate cases are excluded as having no interest.
H being the orthocentre of a triangle ABC, we may call the angles HAC, HBA, HCB = x, y, z respectively, for their sum is 90°.Now tan x tan z = ∴ tan x tan z + tan z tan y + tan x tan y = 1.
Geometrical Proof of tan A + tan B + tan C = tan A tan B tan C