RESEARCH ARTICLE Copyright © 2013 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Advanced Mathematics and Applications Vol. 2, 1–6, 2013 Monotonicity Results Concerning Certain Lengths within a Triangle Toufik Mansour ∗ and Mark Shattuck Department of Mathematics, University of Haifa, 31905 Haifa, Israel We consider several pairs of symmetric parts in a triangle and determine when the congruence of these parts is equivalent to the congruence of the sides in question. We do so by establishing a more general monotonicity property in several cases. Keywords: Altitude, Angle Bisector, Median, Steiner-Lehmus Theorem. 1. INTRODUCTION Two parts of an isosceles triangle which are mirror images with respect to the line of symmetry are always congruent. Many examples of this are given as exercises involving triangle congruence in a first course in geometry. Con- versely, one can ask if congruence of some particular pair of symmetric parts in a triangle implies congruence of two sides. A typical exercise in this regard might be to show that the congruence of medians or altitudes to two sides implies congruence of the sides. That congru- ence of two angle bisectors implies congruence of the sides, on the other hand, is more difficult and is the con- tent of the well known Steiner-Lehmus theorem. Often results such as these are particular cases of more gen- eral monotonic behavior. For example, a median, altitude or angle bisector to a longer side is always shorter, and conversely. See, e.g., (Hall, 2004; Coxeter, 1961, 1967; Hajja, 2008a; Oláh-Gál, 2009) for various proofs of the Steiner-Lehmus theorem and Hajja (2008b) for stronger versions of it. See also Sastry (2005) for a compara- ble result from the Gergonne cevian perspective and Hall (2004) for results involving extensions of the angle bisector. In this paper, we consider a variant of the Steiner- Lehmus problem gotten by looking at certain pairs of symmetric parts within a triangle and determining when congruence of these parts implies the congruence of the relevant sides. We show this by establishing more gen- eral monotonicity. We first look at the four pairs of seg- ments determined by the intersection of two altitudes with two medians and then consider the comparable problems involving altitudes and angle bisectors as well as medians and angle bisectors. ∗ Author to whom correspondence should be addressed. In several of the proofs, we will make use of the fol- lowing result which is known as the Theorem of Menelaus (see, e.g., (Coxeter, 1967, p. 66)). Theorem 1. If ABC is a triangle and D is on an exten- sion of ABE is on side BC , and F is on side AC , then the three points DE, and F are collinear, if and only if,  AD DB  BE EC  CF FA  =-1 (1) Here, the measure of a segment in one direction is consid- ered to be the opposite of its measure in the other. Note that the ratio AD/DB in the figure above is negative since the line DEF (called a transversal) intersects segment AB externally, and thus AD and DB have opposite direction, while the ratios BE EC and CF FA are positive since DEF intersects BC and AC internally. Theorem 1 also applies when the line DEF lies entirely outside of the triangle ABC (in which case all three ratios are negative). In practice, when using the “only if” direc- tion of this theorem, it is sometimes convenient to take the absolute value of both sides of (1) and no longer consider segments as signed. 2. SOME MONOTONICITY RESULTS We show that the congruence of a pair of symmetric parts is equivalent to the congruence of the sides in question in several instances by showing more general monotonicity. 2.1. Altitudes and Medians Our first result concerns the four pairs of segments deter- mined by the intersection of a pair of altitudes with a pair of medians to two given sides. J. Adv. Math. Appl. 2013, Vol. 2, No. 1 2156-7565/2013/2/001/006 doi:10.1166/jama.2013.1026 1