Advanced Geometric Inequalities

1. Introduction

Geometric inequalities ask us to bound distances, areas, perimeters, and angles within a figure — and to determine exactly when those bounds are achieved. They sit at the intersection of synthetic geometry and the great algebraic inequalities (AM–GM, Cauchy–Schwarz), and they reward a solver who can fluidly translate between a picture and an algebraic expression. On the AIME and at the olympiad level, these problems test both creativity (which classical inequality or transformation unlocks the figure?) and rigor (is the extremal configuration actually attainable?).

The defining feature of an inequality problem, as opposed to an equality computation, is the equality case. Half the work is producing the bound; the other half is identifying the configuration where equality holds, because that configuration is usually the extremum the problem is really asking about. Equality in AM–GM means all terms equal; in Cauchy–Schwarz, proportional sequences; in Ptolemy's inequality, a cyclic quadrilateral; in the triangle inequality, collinearity. Memorizing these conditions turns hard optimization problems into "find the configuration that forces equality."

This article assembles the toolkit: the triangle inequality, AM–GM and Cauchy–Schwarz applied to lengths, Ptolemy's theorem and inequality, the Erdős–Mordell inequality, reflection and the shortest-path principle, and the use of loci. Each tool comes with its equality condition and worked contest problems, building from rectangle optimization up to olympiad-level path and quadrilateral problems.

The equality-case principle. In every problem below, the bound is only half the answer. You must also identify when equality holds and confirm that configuration is achievable in the problem's domain. A bound of $L_{\min} = 4\sqrt{2}$ is useless if the point $P$ is constrained to a region where equality is impossible.

Algebraic vs synthetic. AM–GM and Cauchy–Schwarz are algebraic hammers; reflection and Ptolemy are synthetic hammers. When coordinates are already given, go algebraic. When the figure has symmetry, circles, or angle constraints, go synthetic first.

2. Core Concepts

2.1 The Triangle Inequality

For any three points $A, B, C$, $AB + BC \ge AC,$ with equality if and only if $B$ lies on segment $AC$ (the points are collinear in that order). This is the most basic geometric constraint and the foundation of all "shortest path" reasoning: a straight line is the shortest distance between two points, and any detour is at least as long.

2.2 AM–GM Inequality

For nonnegative reals $a_1, \ldots, a_n$, $\frac{a_1 + a_2 + \cdots + a_n}{n} \ge \sqrt[n]{a_1 a_2 \cdots a_n},$ with equality if and only if $a_1 = a_2 = \cdots = a_n$. In geometry it bounds products (areas, volumes) by sums (perimeters, fixed totals), and pins the extremum at the "most symmetric" configuration.

2.3 Cauchy–Schwarz Inequality

For real sequences $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$, $\left(\sum_{i=1}^n a_i b_i\right)^2 \le \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right),$ with equality iff the sequences are proportional, $a_i = \lambda b_i$. The Engel form (Titu's lemma) is especially handy for sums of fractions: $\frac{a_1^2}{b_1} + \frac{a_2^2}{b_2} + \cdots + \frac{a_n^2}{b_n} \ge \frac{(a_1 + a_2 + \cdots + a_n)^2}{b_1 + b_2 + \cdots + b_n}, \qquad b_i > 0.$

2.4 Ptolemy's Theorem and Inequality

For any four points $A, B, C, D$ in the plane, $AC \cdot BD \le AB \cdot CD + AD \cdot BC \qquad \text{(Ptolemy's inequality)},$ with equality if and only if $ABCD$ is a cyclic quadrilateral (the four points lie on a circle, in that order). The equality version is Ptolemy's theorem: in a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides.

2.5 The Erdős–Mordell Inequality

For a point $P$ inside triangle $ABC$, let $p, q, r$ be the distances from $P$ to the three sides. Then $PA + PB + PC \ge 2(p + q + r),$ with equality if and only if $ABC$ is equilateral and $P$ is its center. This deep inequality frequently appears in olympiad shortlists and bounds vertex distances by side distances.

2.6 Reflection and the Shortest-Path Principle

To minimize a broken path that touches a line $\ell$ (e.g. "go from $A$ to a point on $\ell$ then to $B$"), reflect one endpoint across $\ell$. The path length equals the distance from the reflected point to the other endpoint, minimized when the path is straight. This converts an optimization into "draw a straight line," and the optimum occurs where the incoming and outgoing angles with $\ell$ are equal (the reflection/light-ray principle).

2.7 Loci

A locus is the set of points satisfying a geometric condition: points at a fixed distance from a center form a circle; points from which a fixed segment subtends a fixed angle form a circular arc (the inscribed-angle locus); points equidistant from two fixed points form the perpendicular bisector. Recognizing the relevant locus often reveals the extremal point as a tangency or intersection.

2.8 Holder's Inequality and Power Means (Contest Forms)

Holder's inequality generalizes Cauchy–Schwarz: for positive $p, q$ with $\frac1p + \frac1q = 1$, $\sum a_i b_i \le \left(\sum a_i^p\right)^{1/p}\left(\sum b_i^q\right)^{1/q}.$ The power mean inequality states that for positive $a_i$, the $r$-th power mean increases with $r$. For geometry, the special cases AM–GM ($r = 1$ vs $r = 0$) and Cauchy–Schwarz ($r = 2$) suffice.

2.9 Jensen's Inequality for Convex Functions

If $f$ is convex on an interval and $w_i \ge 0$ with $\sum w_i = 1$, then $f\left(\sum w_i x_i\right) \le \sum w_i f(x_i).$ For $f(x) = x^2$, this gives the QM–AM inequality. In geometry, Jensen bounds weighted averages of squared distances.

2.10 Stewart's Theorem and Cevian Lengths

In $\triangle ABC$ with cevian $AD$ to side $BC$ where $BD = m$, $DC = n$, and $AD = d$: $man + dad = bmb + cnc,$ i.e. $b^2 m + c^2 n = a(d^2 + mn)$. Stewart's theorem gives exact cevian lengths, which can then be bounded via AM–GM or Cauchy–Schwarz.

2.11 Weitzenböck's Inequality

For any triangle with area $A$ and side lengths $a, b, c$, $A \le \frac{\sqrt{3}}{12}(a^2 + b^2 + c^2),$ with equality iff the triangle is equilateral. This is a classic area-vs-perimeter-squared bound.

2.12 Fagnano's Problem and the Orthic Triangle

Among all inscribed triangles in an acute triangle, the one with minimum perimeter is the orthic triangle (vertices at the feet of the altitudes). The proof uses double reflection — a flagship application of the reflection principle.

2.13 Minkowski's Inequality for Lengths

In the plane, $\|u + v\| \le \|u\| + \|v\|$ is the triangle inequality on vectors. For sums of Euclidean norms, $\sum_{i=1}^n \|v_i\| \ge \left\|\sum_{i=1}^n v_i\right\|,$ which bounds broken paths by the straight-line distance.

2.14 Isoperimetric Principles in Contest Geometry

Among plane figures of fixed perimeter, the circle encloses maximum area; among $n$-gons of fixed perimeter, the regular $n$-gon is best. Contest versions: fixed perimeter $\Rightarrow$ square maximizes area among rectangles; fixed area $\Rightarrow$ square minimizes perimeter.