New PDF release: An Intro. to Differential Geometry With Applns to Elasticity

By P. Ciarlet

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Then for all points Ρ φ A on the line AD we have \ΔΒΑΡ\° = \ΔΡΑΟ\°. We call AP the mid-line or bisector of the support \BAC. 17. Mid-line of a n a n g l e - s u p p o r t . 5. Any angle £BAC such that 0 < \ZBAC\° < 90 is called acute, such that \ΔΒΑΰ\° = 90 is called right, and such that 90 < \ΔΒΑΟ\° < 180 is called obtuse. If ZBAC is a right-angle, then the lines AB and AC are said to be perpendicular to each other, written AB ± AC. 18. Perpendicular lines. 19. Congruent triangles. Our treatment of congruence If [A, B,C], [Α',Β',Ο'] are triangles such that |B,C7| = |5',C7'|, \C,A\ = \C,A'\, \ZBAC\° = \ΔΒ'Α'0'\\ \A,B\ = \A',B'\, ο \ΔΟΒΑ\° = \Δ0'Β'Α'\ , \ΔΑΟΒ\° = \ΔΑ'ΰ'Β'\°, then we say by way of definition that the triangle [A, B,C]ia congruent to the triangle [A',B',C'].

Now R € ft, C € ft so by A3 (iii) there is some point D of [B,C] on Z, so that D € [ £ , £ ] , £ € Z. 4 we cannot have A e [R,C]. Hence Αφ D. But A € l,D € Ζ so by A AD = I. However AB = BC and D 6 BC, so D € AR. Then AB = AD = I, so R € /. This gives a contradiction. Thus the original supposition is untenable so [A,R] C Hi, and this proves (iii). 3) ANGLE-SUPPORTS, REGIONS, ANGLES 27 (iv) CASE (a). Let Β el. Then [ i , f l C i c W i , which gives the desired conclusion. CASE (b). Let Be ft. Suppose that [Α,Β is not a subset of U\.

In each case the point A is called the v e r t e x of the angle, the half-lines [Α,Β and [A, C are called the a r m s of the angle, and \BAC is called the s u p p o r t of the angle. We denote a wedge-angle with support \BACl by ABAC. The wedge-angle ΔΒΑΒ is said to be a n u l l - a n g l e . 1 TRIANGLES AND CONVEX QUADRILATERALS Terminology COMMENT. The terminology which we have used hitherto is established, apart from 'angle-support' and 'wedge-angle' which we have coined. Now we are reaching termi­ nology which is of long standing but is used in slightly varying senses.

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An Intro. to Differential Geometry With Applns to Elasticity by P. Ciarlet

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