Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

the euclidean diagram (1995) 95APFig. 4.2.Bbecause they are (properly) based on diagram appearance, there is a premiumon controlling this.An example is the striking argument that all triangles are isoceles, popularin discussions of ‘geometrical fallacies’ (Maxwell (1961), Dubnov (1963)). Letthe given triangle be ABC (Fig. 4.3). Let the bisector of angle (BAC) andthe perpendicular bisector (at D) ofthesideBC intersect at O. Drawtheperpendiculars OQ and OR to the remaining sides AC and AB; connect OBand OC. We obtain the following three congruences: (a) ODB ≃ ODC,by side-angle-side; hence OB = OC. (b)AOR congruent to AOQ, by(bisected) angle-(common) side-(complemetary) angle; hence OR = OQ. Butthen (c) BOR congruent to BOQ, because they are right triangles withequal side and hypotenuse; hence RB = QC. Sofar,sogood.It is, however, possible to draw a diagram with AO not quite bisectingthe angle, and DO not quite bisecting the side (Fig. 4.4), so as to displayR and Q either (i) both on AB respectively AC, or (ii) both beyond ABABRDQCFig. 4.3.O