What are the steps for constructing an inscribed circle?

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Concurrence and Constructions
Constructions related to theorems involving circumcenters and other intersection points of three or more lines.
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No center point?
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Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Non-Euclidean constructions
What are the steps for constructing an inscribed circle in triangle ABC only a compass and a straightedge?
What step is needed when constructing a circle inscribed in a triangle?
What is the correct order of constructing a circumscribed circle?
How do you construct the inscribed and circumscribed circles of a triangle?

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Concurrence and Constructions

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Inscribed and Circumscribed Circles of Triangles

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How to construct a square inscribed in a given circle. The construction proceeds as follows:

A diameter of the circle is drawn.
A perpendicular bisector of the diameter is drawn using the method described in Perpendicular bisector of a segment. This is also a diameter of the circle.
The resulting four points on the circle are the vertices of the inscribed square.

No center point?

If the circle's center point is not given, it can be constructed using the method in Constructing the center of a circle.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

	Argument	Reason
1	AC is a diameter of the circle O	A diameter is a line through the circle center. See Diameter definition.
2	BD is a diameter of the circle O	It was drawn using the method in Perpendicular bisector of a line. See that page for proof. The center of a circle bisects the diameter, so BD passes through the center.
3	AC, BD are perpendicular	BD was drawn using the method in Perpendicular bisector of a line. See that page for proof.
4	AC, BD bisect each other	Both are diameters of the circle O. (1), (2) and the center of a circle bisects its diameter. See Diameter definition
5	ABCD is a square	Diagonals of a square bisect each other at 90°. (3), (4)
6	ABCD is an inscribed square	All vertices lie on the given circle O

- Q.E.D

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Other constructions pages on this site

List of printable constructions worksheets

Lines

Introduction to constructions
Copy a line segment
Sum of n line segments
Difference of two line segments
Perpendicular bisector of a line segment
Perpendicular at a point on a line
Perpendicular from a line through a point
Perpendicular from endpoint of a ray
Divide a segment into n equal parts
Parallel line through a point (angle copy)
Parallel line through a point (rhombus)
Parallel line through a point (translation)

Angles

Bisecting an angle
Copy an angle
Construct a 30° angle
Construct a 45° angle
Construct a 60° angle
Construct a 90° angle (right angle)
Sum of n angles
Difference of two angles
Supplementary angle
Complementary angle
Constructing 75° 105° 120° 135° 150° angles and more

Triangles

Copy a triangle
Isosceles triangle, given base and side
Isosceles triangle, given base and altitude
Isosceles triangle, given leg and apex angle
Equilateral triangle
30-60-90 triangle, given the hypotenuse
Triangle, given 3 sides (sss)
Triangle, given one side and adjacent angles (asa)
Triangle, given two angles and non-included side (aas)
Triangle, given two sides and included angle (sas)
Triangle medians
Triangle midsegment
Triangle altitude
Triangle altitude (outside case)

Right triangles

Right Triangle, given one leg and hypotenuse (HL)
Right Triangle, given both legs (LL)
Right Triangle, given hypotenuse and one angle (HA)
Right Triangle, given one leg and one angle (LA)

Triangle Centers

Triangle incenter
Triangle circumcenter
Triangle orthocenter
Triangle centroid

Circles, Arcs and Ellipses

Finding the center of a circle
Circle given 3 points
Tangent at a point on the circle
Tangents through an external point
Tangents to two circles (external)
Tangents to two circles (internal)
Incircle of a triangle
Focus points of a given ellipse
Circumcircle of a triangle

Polygons

Square given one side
Square inscribed in a circle
Hexagon given one side
Hexagon inscribed in a given circle
Pentagon inscribed in a given circle

Non-Euclidean constructions

Construct an ellipse with string and pins
Find the center of a circle with any right-angled object

What are the steps for constructing an inscribed circle in triangle ABC only a compass and a straightedge?

Construct an Inscribed Circle Draw a triangle. Use your compass and straightedge to construct the angle bisector of one of the angles. Repeat with a second angle. The point of intersection of the angle bisectors is the incenter.

What step is needed when constructing a circle inscribed in a triangle?

Construct the angle bisectors of each angle in the triangle. Construct the perpendicular bisectors of each side of the triangle. Place the compass on a vertex and use the bisectors to draw the circle. Place the compass on the intersection of the bisectors to draw the circle.

What is the correct order of constructing a circumscribed circle?

Construct the perpendicular bisector of one side of triangle. Construct the perpendicular bisector of another side. Where they cross is the center of the Circumscribed circle. Place compass on the center point, adjust its length to reach any corner of the triangle, and draw your Circumscribed circle!