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6.R. GEOMETRIC FALLACIES
General surveys of such fallacies can be found in the following. See also: 6.P, 10.A.1.

These fallacies are actually quite profound as the first two point out some major gaps in Euclid's axioms -- the idea of a point being inside a triangle really requires notions of order of points on a line and even the idea of continuity, i.e. the idea of real numbers.

Ball. MRE. 1st ed., 1892, pp. 31 34, two examples, discussed below. 3rd ed., 1896, pp. 39 46 = 4th ed., 1905, pp. 41-48, seven examples. 5th ed., 1911, pp. 44-52 = 11th ed., 1939, pp. 76-84, nine example.

Walther Lietzmann. Wo steckt der Fehler? Teubner, Stuttgart, (1950), 3rd ed., 1953. (Strens/Guy has 3rd ed., 1963(?).) (There are 2nd ed, 1952??; 5th ed, 1969; 6th ed, 1972. MG 54 (1970) 182 says the 5th ed appears to be unchanged from the 3rd ed.) Chap. B: V, pp. 87-99 has 18 examples.

(An earlier version of the book, by Lietzmann & Trier, appeared in 1913, with 2nd ed. in 1917. The 3rd ed. of 1923 was divided into two books: Wo Steckt der Fehler? and Trugschlüsse. There was a 4th ed. in 1937. The relevant material would be in Trugschlüsse, but I have not seen any of the relevant books, though E. P. Northrop cites Lietzmann, 1923, three times -- ??NYS.)

E. P. Northrop. Riddles in Mathematics. 1944. Chap. 6, 1944: 97-116, 232-236 & 249-250; 1945: 91-109, 215-219 & 230-231; 1961: 98-115, 216-219 & 229. Cites Ball, Lietzmann (1923), and some other individual items.

V. M. Bradis, V. L. Minkovskii & A. K. Kharcheva. Lapses in Mathematical Reasoning. (As: [Oshibki v Matematicheskikh Rassuzhdeniyakh], 2nd ed, Uchpedgiz, Moscow, 1959.) Translated by J. J. Schorr-Kon, ed. by E. A. Maxwell. Pergamon & Macmillan, NY, 1963. Chap. IV, pp. 123-176. 20 examples plus six discussions of other examples.

Edwin Arthur Maxwell. Fallacies in Mathematics. CUP, (1959), 3rd ptg., 1969. Chaps. II-V, pp. 13-36, are on geometric fallacies.

Ya. S. Dubnov. Mistakes in Geometric Proofs. (2nd ed., Moscow?, 1955). Translated by Alfred K. Henn & Olga A. Titelbaum. Heath, 1963. Chap 1-2, pp. 5-33. 10 examples.

А. Г. Конфорович. [A. G. Konforovich]. (Математичні Софізми і Парадокси [Matematichnī Sofīzmi ī Paradoksi] (In Ukrainian). Радянська Школа [Radyans'ka Shkola], Kiev, 1983.) Translated into German by Galina & Holger Stephan as: Konforowitsch, Andrej Grigorjewitsch; Logischen Katastrophen auf der Spur – Mathematische Sophismen und Paradoxa; Fachbuchverlag, Leipzig, 1990. Chap. 4: Geometrie, pp. 102-189 has 68 examples, ranging from the type considered here up through fractals and pathological curves.

S. L. Tabachnikov. Errors in geometrical proofs. Quantum 9:2 (Nov/Dec 1998) 37-39 & 49. Shows: every triangle is isosceles (6.R.1); the sum of the angles of a triangle is 180^o without use of the parallel postulate; a rectangle inscribed in a square is a square; certain approaching lines never meet (6.R.3); all circles have the same circumference (cf Aristotle's Wheel Paradox in 10.A.1); the circumference of a wheel is twice its radius; the area of a sphere of radius R is π²R².