Geometric Constructions: Circles, Polygons, Conics and Ovals

Geometric Constructions with Compass and Straightedge

Perpendicular Bisectors and Midpoints

Draw the medians joining the midpoints of the sides of a triangle; they meet at the centroid (G). Draw the perpendicular bisectors (mediatrices) of each side of a triangle. Their intersection is the circumcenter (O), the center of the circumscribed circle.

Angle Bisectors and the Incircle

Draw the angle bisectors. Their intersection is the incenter (I), the center of the inscribed circle.

Regular Polygon Construction

Given a regular pentagon with known side AB: transfer segment AB and erect the perpendicular on side AB. Draw the perpendicular bisector. Transfer the radius, mark the points and connect them to complete the polygon.

Given a regular polygon with a known circumradius: draw the vertical diameter and divide it into N equal parts. With radius equal to the diameter, draw arcs from the endpoints of the diameter, then join the points to form the polygon.

Given a regular polygon with a known side: transfer the side AB, draw the perpendicular bisector, and construct the center O with radius OB to complete the figure.

Tangent Circles

To draw external and internal tangent circles to two given circles: join the centers O1 and O2 and draw the perpendicular bisector. Add or subtract the radii as required. Draw a circle with center O1 and radius O1-P, then join P and Q. Draw a parallel to the line through the centers to locate the tangent points T1 and T2.

To draw a circle tangent to a given line and passing through a point: locate the point outside the circle, then construct the required circle using the perpendicular bisector of the segment from the point to the line.

To draw a circle tangent to two given circles: use centers O1 and O2 and radii r1 and r, adding or subtracting radii as needed, and join the resulting centers with O.

Joining Two Straight Lines with an Arc

To connect two lines using a known arc, draw the bisector of the angle formed, locate the center O1 at the required radius, and draw the connecting arc between the tangent points T1 and T2.

Ovals

Oval from a known side: divide AB into three parts (Thales), construct auxiliary circles, and join the centers O1, O2, O3 and O4 to obtain the tangent points that form the oval.

Oval from a known diameter CD: take a point O1, draw arcs with radius equal to half of CD, and draw the vertical diameter O3O4. Connect the centers to form the four arcs of the oval.

Oval from its two axes: transfer AB and CD, draw the bisector of AQ, and join the centers O1, O2, O3 and O4 to draw the arcs.

Oval inscribed in a rhombus: construct the rhombus, draw the perpendicular sides from the opposite vertices, and draw the arcs centered at the vertices O1, O2, O3 and O4.

Ovoid from a known axis AB (6 parts): divide AB into parts, draw circles centered at O1 and O2, draw the vertical diameter, and join O2 with O3 and O4 to draw the arcs.

Conic Sections

Ellipse: with foci F1 and F2 located at half the distance of the major axis AB, draw arcs from F1 and F2 with radius equal to the major axis. Points on the ellipse are the intersections of these arcs.

Hyperbola: take points outside the foci F1 and F2. Using the vertices V1 and V2, draw arcs from the foci to obtain points on the curve. Draw the tangent lines at the desired points.

Parabola: take points on the axis from the vertex V. Draw perpendiculars to the axis through these points, transfer distances from the focus, and draw the tangent lines.

Isometric Projection

Draw the isometric axes at 30° angles. Set the bounding box dimensions (for example, 7 mm × 5 mm). Draw circles with diameter 80 and apply the isometric ratio 0.816 to construct the isometric ellipses. Connect the midpoints of the axes to draw the four sides of the oval arcs.