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THEOREMS IN ELEMENTARY GEOMETRY

IP.com Disclosure Number: IPCOM000128827D
Original Publication Date: 1999-Sep-11
Included in the Prior Art Database: 2005-Sep-19
Document File: 2 page(s) / 13K

Publishing Venue

Software Patent Institute

Related People

M. L. Perez: AUTHOR [+3]

Abstract

1) Smarandache Concurrent Lines 2) Smarandache Cevians Theorem 3) Smarandache Podaire Theorem 4) Generalization of the Bisector Theorem 5) Generalization of the Altitude Theorem 6) Collinear Points 7) Median Point Theorem

This text was extracted from a PDF file.
This is the abbreviated version, containing approximately 53% of the total text.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Copyright M.L. Perez, 1999 All Rights Reserved

THEOREMS IN ELEMENTARY GEOMETRY

edited by M. L. Perez

1) Smarandache Concurrent Lines

If a polygon with n sides (n = 4) is circumscribed to a circle, then there are at least three concurrent lines among the polygon's diagonals and the lines which join tangential points of two non-adjacent sides. (This generalizes a geometric theorem of Newton.)

Reference: F. Smarandache, "Problemes avec and sans problemes!" (French: Problems with and without ... Problems!), Ed. Somipress, Fes, Morocco, 1983, Problem & Solution # 5.36, p.
54.

2) Smarandache Cevians Theorem

Let AA', BB', CC' be three concurrent cevians (lines) in the point P in the triangle ABC. Then:

PA/PA' + PB/PB' + PC/PC' = 6, and PA PB PC BA CB AC ---- . ---- . ---- = ---- . ---- . ---- = 8.

PA' PB' PC' BA' CB' AC'

Reference: F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress, Fes, Morocco, 1983, Problems & Solutions # 5.37, p. 55, # 5.40, p. 58.

3) Smarandache Podaire Theorem

Let AA', BB', CC' be the altitudes (heights) of the triangle ABC.

Thus A'B'C' is the podaire triangle of the triangle ABC.

Note AB = c, BC = a, CA = b, and A'B' = c', B'C' = a', C'A' = b'.

Then: a'b' + b'c' + c'a' <= 1/4 (a^2 + b^2 + c^2) Reference: F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress, Fes, Morocco, 1983, Problem & Solution # 5.41, p. 59.

4) Generalization of the Bisector Theorem

Let AM be a cevian of the triangle ABC which forms the angles A1 and A2 with the sides AB an...