Angle 54° Trisection

8.2 Angle 54° Trisection
Level Information
Instruction:	Construct two rays that divide the given angle of 54° into three equal parts.
Goal:	4L 5E
Available Tools:
Location Information
Pack:	Theta
Previous Level:	8.1 Perimeter Bisector
Next Level:	8.3 Interior Angles

4L solution

Let A be the vertex.

1. Construct a circle with center B on either line and radius AB. Circle B intersects with line AB at point C, and the other line at point D.
2. Construct a circle with center D and radius AD. Circle D intersects with circle B at point E.
3. Draw line AE, the first angle trisector.
4. Bisect angle DAE, the second angle trisector.

Alternative 4L solution:

1. Construct a circle with center B on either line and radius AB, intersecting the other line at point C.
2. Constuct the parallel to AB through C, intersecting circle B at point D.
3. Constuct line AD, the first angle trisector.
4. Construct the angle bisector of ∠DAB, the second angle trisector.

5E solution

Let A be the point of intersection between the two lines

1. Construct a circle with center B on either line and radius AB. Circle B intersects with line AB at point C, and the other line at point D.
2. Construct a circle with center D and radius AD. Circle D intersects with circle B at point E.
3. Draw line AE. This is the first angle trisector.
4. Draw circle with center E and radius EC. Circle E intersects with circle B at point F.
5. Draw line AF. This is the second angle trisector.

Explanation

As mentioned in the FAQ, trisection of an angle is not possible in general case, so you need to use that 54/3 = 18. You can also note that 90-54 = 36 = 2*18 and so on.

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