Elegant Geometric Constructions
Paul Yiu
Department of Mathematics
Florida Atlantic University

[pdf]

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.