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Square Minimization

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April 6, 2008
geometryanalytic geometrygraphing linesslopetrigonometryinequalitiesquadratics

Problem Statement

Let ABC ABC be a triangle with AB \equal{} 5, BC \equal{} 4 and AC \equal{} 3. Let P \mathcal P and Q \mathcal Q be squares inside ABC ABC with disjoint interiors such that they both have one side lying on AB AB. Also, the two squares each have an edge lying on a common line perpendicular to AB AB, and P \mathcal P has one vertex on AC AC and Q \mathcal Q has one vertex on BC BC. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph;
unitsize(1.5cm);
pair A, B, C;
A = origin; B = A + 5 * right; C = (9/5, 12/5);
pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45);
pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C);
pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2;
pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y);
draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle);
label("AA", A, W); label("BB", B, E); label("CC", C, N);
label("P\mathcal P", Oa, origin); label("Q\mathcal Q", Ob, origin);[/asy]
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