Heron's Formula and Ptolemy's Theorem
نویسنده
چکیده
We adopt the following rules: p1, p2, p3, p4, p5, p6, p, p7 denote points of E2 T and a, b, c, r, s denote real numbers. Next we state four propositions: (1) If sin](p1, p2, p3) = sin](p4, p5, p6) and cos](p1, p2, p3) = cos](p4, p5, p6), then ](p1, p2, p3) = ](p4, p5, p6). (2) sin](p1, p2, p3) = −sin](p3, p2, p1). (3) cos](p1, p2, p3) = cos](p3, p2, p1). (4) ](p1, p4, p2)+](p2, p4, p3) = ](p1, p4, p3) or ](p1, p4, p2)+](p2, p4, p3) = ](p1, p4, p3) + 2 · π. Let us consider p1, p2, p3. The area of M(p1, p2, p3) yields a real number and is defined by: (Def. 1) The area of M(p1, p2, p3) = 2 · (((p1)1 · (p2)2 − (p2)1 · (p1)2) + ((p2)1 · (p3)2 − (p3)1 · (p2)2) + ((p3)1 · (p1)2 − (p1)1 · (p3)2)).
منابع مشابه
Ptolemy's Theorem
This entry provides an analytic proof to Ptolemy’s Theorem using polar form transformation and trigonometric identities. In this formalization, we use ideas from John Harrison’s HOL Light formalization [1] and the proof sketch on the Wikipedia entry of Ptolemy’s Theorem [3]. This theorem is the 95th theorem of the Top 100 Theorems list [2].
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ورودعنوان ژورنال:
- Formalized Mathematics
دوره 16 شماره
صفحات -
تاریخ انتشار 2008