Elegant Geometric Constructions
Paul Yiu
Department of Mathematics
Florida Atlantic University

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 Paragraph Construction [GSP files] 2.1 Construction of regular octagon: to construction a regular octagon by cutting corners of a square. 2.2 Given two congruent circles each passing through the center of the other, to construct a circle tangent internally to one, extenrally to the other, and also the line joining their centers. 2.3 Construction 1: Equilateral triangle inscribed in a rectangle: given a rectangle ABCD, to construct two points P on BC and Q on CD such that triangle APQ is equilateral. 2.4 Partition of equilateral triangle with four congruent incircles. 3.1 Construction 2: Geomeric mean. 3.1 Solution of quadratic equations x(x-a) = b^2 and x(a-x) = b^2. 3.1.1 Construction 3: Given a chord BC perpendicular to a diameter XY of a circle, to construct a point A on the arc BYC such that AX intersects BC at T with AT equal to a given length. 3.2 Harmonic mean  and  the  equation  1/a  +  1/b = 1/x. 4.1 Construction 4: Archimedes' twin circles in the shoemaker's knife. 4.1 Constructions 5 - 6: Incircles of shoemaker's knife. 4.2 Constructions 7 - 10: Four very simple constructions of the incircle of a shoemaker's knife. 5.1 Euler's formula for the distance between circumcenter and incenter of a triangle. 5.1.1 Construction 11: Given the circumcircle (O) and the inradius r of a triangle, to construct the incircle. 5.1.2 Construction 12: Given the circumcircle (O) and the incenter I of a triangle, to construct the incircle. 5.1.3 Construction 13: Given the incircle (I) and circumcenter O of a triangle, to construct the circumcircle. 5.1.4 Construction 14: Given the incircle (I) and circumradius R of a triangle, to construct the circumcircle. 5.2.1 Construction 15: Given I in (O), to construct (I) which is the incircle of quadrilaterals inscribed in (O). 5.2.2 Construction 16: Given (O) and r, to construct (I) which has radius r and is the incircle of quadrilaterals inscribed in (O). 5.2.3 Construction 17: Given (I) and O, to construct (O) which is the circumcircle of quadrilaterals with incircle (I). 6.1 Construction 18: Given P on a chord BC of a circle (O), to construct the two circles tangent to (O) and to a chord BC at P. 6.2 Construction 19: The two neighbors of a circle tangent to (O) and to a chord BC at P. 6.3 Construction 20: Mixtilinear incircle. 6.4 Construction 21: Ajima construction of circle tangent to the circumcircle of ABC and to PB, PC. 6.4.1 Thébault's theorem. 6.4.2 An animation picture of squares and circles. 7.1 Construction 22: To construct a triangle given the length of one side, and the median and angle bisector on that side. 7.2 Construction 23: To construct a triangle given an angle and the corresponding median and angle bisector. 7.3 Construction 24: To construct a triangle given a vertex, incenter, and orthocenter.

Last update: February 8, 2005.