Scientia Magna Vol. 6 (2010), No. 2, 26-28
Smarandache’s Orthic Theorem Edited by Ion Patrascu Fratii Buzesti College, Craiova, Romania Abstract In this paper We present the Smarandache’s Orthic Theorem in the geometry of the triangle. Keywords Smarandache’s Orthic Theorem, triangle.
§1. The main result Smarandache’s Orthic Theorem Given a triangle ABC whose angles are all acute (acute triangle), we consider A0 B 0 C 0 , the triangle formed by the legs of its altitudes. In which conditions the expression: kA0 B 0 k · kB 0 C 0 k + kB 0 C 0 k · kC 0 A0 k + kC 0 A0 k · kA0 B 0 k is maximum?
A b−y
z C c−z B
x
B
0
0
y 0
A
a−x
C
Proof. We have 4ABC ∼ 4A0 B 0 C 0 4AB 0 C ∼ 4A0 BC 0 . We note kBA0 k = x, kCB 0 k = y, kAC 0 k = z. It results that kA0 Ck = a − x, kB 0 Ak = b − y, kC 0 Bk = c − z. 0 A0 C = BA 0 C 0 ; ABC 0 C 0 = A\ 0 B 0 C 0 ; BCA 0 A0 = B 0 C 0 A. \ = B\ \ \ = AB \ \ = BC \ \ BAC
From these equalities it results the relation (1)
(1)