Constructing Special Triangles: Isosceles, Right, Right Isosceles, and
Equilateral
Perpendicular Bisectors
Construct a perpendicular bisector of a given line segment.
Step 0: With a straight edge, construct a segment.
Step 1: With compass, construct two congruent circles with the centers at the endpoints of the segment.
Remember the important part of the circles is where they intersect.
Step 2: With the straight edge, construct a line through the two intersections.
Isosceles Triangle
Step 1: With a straight edge, construct an angle.
Step 2: With compass, construct an arc which intersects the sides of the angle.
Step 3: With a straight edge, construct the segment connecting the intersections.
Right Triangle
Step 1: With a straight edge, construct a segment.
Step 2: With compass and straight edge, construct the perpendicular bisector of the segment.
Step 3: Construct a point on the perpendicular bisector. With a straight edge, construct the segment
connecting the point with one of the endpoints of the segment.
Isosceles Right Triangle
Step 1: With a straight edge, construct a segment.
Step 2: With compass and straight edge, construct the perpendicular bisector of the segment.
Step 3: With compass, construct a circle with the midpoint as the center.
Step4: With straight edge, construct segment connecting the intersection of the circle and perpendicular
bisector to the intersection of the circle with the segment.
Equilateral Triangle
Step 1: With a compass, construct a circle.
Step 2: With compass, construct a congruent circle which has as its center a point on the first circle.
Step 3: With a straight edge, construct a triangle with vertices at the two centers and one of the intersections
of the circle.
Constructing Special Segments of a Triangle
Perpendicular Bisectors & Circumcenters
Construct a perpendicular bisector of a given line segment.
Step 0: With a straight edge, construct a segment.
Step 1: With compass, construct two congruent circles with the centers at the endpoints of the segment.
Remember the important part of the circles is where they intersect.
Step 2: With the straight edge, construct a line through the two intersections.
Construct the circumcenter of a triangle.
Step 0: With a straight edge, construct a triangle.
Step 1: With compass and straight edge, construct the perpendicular bisectors of the three sides of the
triangle.
Step 2: Construct a point where they intersect.
Medians & Centroids
Construct the median of a triangle.
Step 0: With a straight edge construct a triangle.
Step 1: With compass and straight edge, construct the perpendicular bisector of one of the sides. Construct a
point at the midpoint.
Step 2: With straight edge, construct the segment from the midpoint to the opposite vertex.
Construct the centroid of a triangle.
Step 0: With a straight edge, construct a triangle.
Step 1: With compass and straight edge, construct the medians of the three sides of the triangle.
Step 2: Construct a point where they intersect.
Altitudes & Orthocenters
Construct a line perpendicular to a given line.
Step 0: With a straight edge, construct a line.
Step 1: Construct a point not on the line. With a compass construct a circle with the point as the center so that
it intersects the line. Remember the important part of the circle is where it intersects the line. Construct
a point at the intersections.
Step 2: With compass and straight edge, construct the perpendicular bisector of the segment which has the
intersections as endpoints.
Construct the shortest distance from a point to a line.
Step 0: With a straight edge, construct a line. Construct a point not on the line.
Step 1: With a compass construct a circle with the point as the center so that it intersects the line. Remember
the important part of the circle is where it intersects the line. Mark the intersections.
Step 2: With compass and straight edge, construct the perpendicular bisector of the segment which has the
intersections as endpoints.
Step 3: Construct with a straight edge and a highlighter the segment from the midpoint of the segment to the
point not on the line.
Construct the altitude of a triangle. (Mimic the construction of the Shortest Distance).
Step 0: With a straight edge, construct a triangle.
Step 1: With a compass construct a circle with a vertex of the triangle as the center so that it intersects the
opposite. Remember the important part of the circle is where it intersects the side. (You may have to
extend the side for one or both intersections). Mark the intersections.
Step 2: With compass and straight edge, construct the perpendicular bisector of the segment which has the
intersections as endpoints. The perpendicular bisector of the segment should be collinear with the
vertex.
Step 3: Construct with a straight edge and a highlighter the segment from the midpoint of the segment to the
vertex opposite the side.
Construct the orthocenter of a triangle.
Step 0: With a straight edge, construct a triangle.
Step 1: With compass and straight edge, construct the altitudes of the three sides of the triangle.
Step 2: Construct a point where they intersect.
Angle Bisectors & Incenters
Construct the bisector of an angle.
Step 0: With a straight edge construct an angle.
Step 1: With compass, construct a circle with the center at the vertex.
Step 2: With compass, construct two congruent circles with the intersections of the circle with the sides of the
angles as the centers.
Step 3: With straight edge, construct the segment through the intersections of the two congruent circles
constructed in Step 2. This segment should be collinear with the vertex.
Construct the Incenter of a triangle.
Step 0: With a straight edge, construct a triangle.
Step 1: With compass and straight edge, construct the angle bisectors of the three angles of the triangle.
Step 2: Construct a point where they intersect.