Square by Two Vertices

The essential idea or method is always grasped intuitively long before any rational argument for the conclusion is devised. — Morris Kline

4.11 Square by Two Vertices
(Mobile app only)
Level Information
Instruction:	Given two vertices of a square. Construct the two other vertices using only a compass.
Goal:	7L 7E 3V
Available Tools:
Location Information
Pack:	Delta
Previous Level:	4.10 Square by Adjacent Midpoints
Next Level:	5.1 Parallel Line

7L 7E solution

Let A and B be the given vertices.

1. Construct the circle with center A and radius AB
2. Construct the circle with center B and radius AB, intersecting circle A at C and D
3. Construct the circle with center C and radius CD, intersecting circle A at E and circle B at F
4. Construct the circle with center E and radius BE
5. Construct the circle with center F and radius AF, intersecting circle E at G and H
6. Construct the circle with center A and radius AG, intersecting circle B at I and J, with I on the same side as G from AB
7. Construct the circle with center B and radius BG, intersecting circle A radius AB at K and L, with K on the same side as I from AB
8. Point at I
9. Point at K

3V solution

Starting from the 7L solution:

1. Point at J
2. Point at L
3. Construct the circle with center G and radius GI, intersecting circle A at M and circle B at N
4. Construct the circle with center K and radius KM
5. Construct the circle with center I and radius IN, intersecting circle K at O
6. Construct the circle with center A and radius AO
7. Construct the circle with center B and radius BO, intersecting circle A radius AO at P
8. Point at P

Explanation

Since there are a large number of points involved, it may be easiest to map points to a coordinate system, by using the Pythagorean theorem on right triangles. Start with A at (0,0) and B at (1,0); the midpoint between A and B would be X at (1/2,0). The first two circles at A and B have radius 1, and construct C at (1/2,√3/2) and D at (1/2,-√3/2) because ABC and ABD are equilateral triangles. Then circle C has radius √3, and constructs E at (-1,0) and F at (2,0) (CEX is a 30-60-90 right triangle; we know leg CX is √3/2 and hypotenuse CE is √3, leaving leg EX at 3/2). Circles E and F each have radius 2, and construct G at (1/2,√7/2) and H at (1/2,-√7/2) (EGX is a right triangle with hypotenuse EG 2 and leg EX 3/2, leaving GX to be √7/2). At this point, we have triangle AGX with legs 1/2 and √7/2, which makes hypotenuse AG √2. So the second circles through A and B form 45-45-90 right triangles ABI, ABJ, ABK, and ABL, putting I at (1,1), J at (1,-1), K at (0,1), and L at (0,-1) giving the points needed for the first two squares (where A and B are adjacent vertices, side length 1).

The remaining variant for the 3V solution uses A and B as opposite rather than adjacent vertices (side length √2/2). With some effort, it can be shown that the triangles AGK and KIM are similar, so the ratio AG:AK = √2:1 = IK:KM = 1:KM, so the circle at K has radius KM = √2/2, likewise for the circle at I. From there, it follows that point O lies at (1/2,1/2) at the midpoint of AI and BK, and the final two circles through A and B reflect point O over AB to point P at (1/2,-1/2).

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