Circumscribed Equilateral Triangle

Logic is the anatomy of thought. — John Locke

4.3 Circumscribed Equilateral Triangle
Level Information
Instruction:	Construct an equilateral triangle that is circumscribed about the circle and contains the given point.
Goal:	5L 6E
Available Tools:
Location Information
Pack:	Delta
Previous Level:	4.2 Angle of 60° - 2
Next Level:	4.4 Equilateral Triangle in Circle

5L solution

Let O be the center of the circle and A the point on it.

1. Construct line OA, intersecting the circle at B
2. Construct the perpendicular to OA from A
3. Construct the circle with center B and radius OB, intersecting the circle at C and D, and line OA at E
4. Construct line CE
5. Construct line DE

6E solution

Let O be the center of the circle and A the point on it.

1. Construct line OA, intersecting the circle at B
2. Construct the circle with center B and radius OB, intersecting the circle at C and D, and line OA at E
3. Construct the circle with center O and radius OE
4. Construct line CE, intersecting the latest circle at F
5. Construct line DE, intersecting the latest circle at G
6. Construct line FG

Explanation

Line OA forms an altitude of the equilateral triangle, with A at the midpoint of the base. Both constructions use circle B to determine the two sides at a 30° angle from the apex; this can be seen since triangles BCO and BDO are equilateral by construction, and angles ECO and EDO are 90 by Thales' theorem. Having drawn tangents through C and D, point E is the apex of the desired triangle. The final leg of the triangle through A is built with the perpendicular tool for the 5L solution, and by inscribing an equilateral triangle into the second larger circle O in the 6E solution.

List of Levels

Alpha Angle of 60° • Perpendicular Bisector • Midpoint • Circle in Square • Rhombus in Rectangle • Circle Center • Inscribed Square Beta Angle Bisector • Intersection of Angle Bisectors • Angle of 30° • Double Angle • Cut Rectangle • Drop a Perpendicular • Erect a Perpendicular • Tangent to Circle at Point • Circle Tangent to Line • Circle in Rhombus Gamma Chord Midpoint • Triangle by Angle and Orthocenter • Intersection of Perpendicular Bisectors • Three equal segments - 1 • Circle through Point Tangent to Line • Midpoints of Trapezoid Bases • Angle of 45° • Lozenge • Center of Quadrilateral Delta Double Segment • Angle of 60° - 2 • Circumscribed Equilateral Triangle • Equilateral Triangle in Circle • Cut Two Rectangles • Square Root of 2 • Square Root of 3 • Angle of 15° • Square by Opposite Midpoints • Square by Adjacent Midpoints • Square by Two Vertices Epsilon Parallel Line • Parallelogram by Three Vertices • Line Equidistant from Two Points - 1 • Line Equidistant from Two Points - 2 • Hash • Shift Angle • Line Equidistant from Two Lines • Circumscribed Square • Square in Square • Circle Tangent to Square Side • Regular Hexagon Zeta Point Reflection • Reflection • Copy Segment • Given Angle Bisector • Non-collapsing Compass • Translate Segment • Triangle by Three Sides • Parallelogram • Nine Point Circle • Symmetry of Four Lines • Parallelogram by Three Midpoints Eta Sum of Areas of Squares • Annulus • Angle of 75° • Line Equidistant from Three Points • Heron's Problem • Circumscribed Circle • Inscribed Circle • Circle Tangent to Three Lines • Segment by Midpoint • Angle Isosceles • Excircle Theta Perimeter Bisector • Angle 54° Trisection • Interior Angles • Regular Octagon • Triangle Cleaver • Torricelli Point • Circle Equidistant from Four Points • Octagon from Square • Egyptian Triangle by Side of Length 4 • Chord Parallel to Segment Iota Minimum Perimeter - 1 • Third Proportional • Harmonic Mean of Trapezoid Bases • Drop a Perpendicular* • Midpoint* • Trisection by Trapezoid Diagonals • Minimum Perimeter - 2 • Harmonic Mean of Segments • Triangle by Angle and Centroid • Triangle Mid-Segment Kappa Tangent of Circle • Outer Tangent • Inner Tangent • Rotation 90° • Rotation 60° • Segment Trisection • Segment Trisection* • Chord Trisection • Three Circles - 1 • Secant Bisection • Three Circles - 2 • Center of Rotation Lambda Fourth Proportional • Geometric Mean of Segments • Golden Section • Angle of 54° • Third Parallel Line • Circle in Angle • Geometric Mean of Trapezoid Bases • Regular Pentagon • Point Farthest from Angle Sides • Ratio 1 to 5 Mu Triangle by Midpoints • Triangle by Side and Centroid • Triangle by Altitude Base Points • Triangle by Tangent Points • Triangle by Excenters • Equilateral Triangle by Centroid and Two Points • Right Triangle by Two Points on Legs • Hypotenuse and Altitude • Hypotenuse and Leg • Isosceles Triangle by Tangent Points Nu Circle Tangent to Line and Circle • Equilateral Triangle - 2 • Equilateral Triangle on Concentric Circles • Square in Triangle • Point Equidistant from Side of Angle and Point • Circle Through Two Points and Tangent to Line • Inscribed Square - 2 • Line Reflection • Square by Four Points Xi Rhombus in Triangle • Circle Tangent to Two Circles • Triangle by Tangent Point on Hypotenuse • Parallelogram on Four Lines • Arbelos Omicron Midpoint* • Copy Circle • Line-Circle Intersection • Three Equal Segments • Tangent to Circle* • Napoleon's Problem • Drop a Perpendicular** • Line-Circle Intersection* • Circle with Center on Line • Angle of 3° • Mickey Mouse Tutorials Equilateral Triangle • Intersect Tool • Perpendicular Bisector • Move Tool • Angle Bisector • Perpendicular • Parallel Line • Compass