By P. Ciarlet
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Extra resources for An Intro. to Differential Geometry With Applns to Elasticity
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.
An Intro. to Differential Geometry With Applns to Elasticity by P. Ciarlet