How to Draw a Circle Inscribed in a Triangle

This folio shows how to construct (draw) an equilateral triangle inscribed in a circle with a compass and straightedge or ruler. This is the largest equilateral triangle that will fit in the circle, with each vertex touching the circle. This is very similar to the construction of an inscribed hexagon, except we utilize every other vertex instead of all six.

Equally can be seen in Definition of a Hexagon, each side of a regular hexagon is equal to the altitude from the center to any vertex. This structure only sets the compass width to that radius, then steps that length off around the circle to create the six vertices of a hexagon.

But instead of drawing a hexagon, we use every other vertex to make a triangle instead. Since the hexagon structure effectively divided the circle into six equal arcs, by using every other indicate, we divide it into three equal arcs instead. The three chords of these arcs form the desired equilateral triangle.

Another fashion of thinking about it is that both the hexagon and equilateral triangle are regular polygons, one with double the number of sides of the other.

The image below is the final drawing from the above animation, simply with extra lines and the vertices labelled.

	Argument	Reason
NOTE: Steps 1 through vii are the same as for the construction of a hexagon inscribed in a circumvolve. In the case of an inscribed equilateral triangle, nosotros use every other betoken on the circumvolve.
1	A,B,C,D,E,F all lie on the circle centre O	By construction.
ii	AB = BC = CD = DE = EF	They were all drawn with the same compass width.
From (2) nosotros meet that five sides are equal in length, but the last side FA was not drawn with the compasses. It was the "left over" space as we stepped around the circle and stopped at F. So nosotros accept to prove it is coinciding with the other five sides.
3	OAB is an equilateral triangle	AB was drawn with compass width set to OA, and OA = OB (both radii of the circle).
iv	grand∠AOB = 60°	All interior angles of an equilateral triangle are threescore°.
five	m∠AOF = 60°	As in (four) chiliad∠BOC, grand∠COD, 1000∠DOE, yard∠EOF are all &60deg; Since all the central angles add to 360°, grand∠AOF = 360 - 5(60)
half dozen	Triangle BOA, AOF are coinciding	SAS See Exam for congruence, side-bending-side.
vii	AF = AB	CPCTC - Corresponding Parts of Coinciding Triangles are Coinciding
And so now we tin evidence that BDF is an equilateral triangle
8	All six primal angles (∠AOB, ∠BOC, ∠COD, ∠DOE, ∠EOF, ∠FOA) are congruent	From (iv) and by repetition for the other 5 angles, all six angles have a measure of 60°
9	The angles ∠BOD, ∠DOF, ∠BOF are congruent	From (8) - They are each the sum of two threescore° angles
10	Triangles BOD, DOF and BOF are congruent.	The sides are all equal radii of the circle, and from (9), the included angles are congruent. See Test for congruence, side-angle-side
11	BDF is an equilateral triangle.	From (ten) BD, DF, FB a re congruent. CPCTC - Respective Parts of Congruent Triangles are Congruent. This in turn satisfies the definition of an equilateral triangle.
12	BDF is an equilateral triangle inscribed in the given circumvolve	From (eleven) and all three vertices B,D,F prevarication on the given circle.

- Q.Eastward.D

Try it yourself

Click here for a printable worksheet containing two problems to try. When you get to the page, use the browser impress command to print as many as you wish. The printed output is not copyright.

Other constructions pages on this site

• Listing 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 north equal parts
• Parallel line through a point (angle re-create)
• Parallel line through a bespeak (rhombus)
• Parallel line through a point (translation)

Angles

• Bisecting an bending
• Re-create an angle
• Construct a 30° angle
• Construct a 45° bending
• Construct a threescore° angle
• Construct a 90° bending (correct angle)
• Sum of n angles
• Deviation of two angles
• Supplementary angle
• Complementary angle
• Constructing  75°  105°  120°  135°  150° angles and more than

Triangles

• Copy a triangle
• Isosceles triangle, given base and side
• Isosceles triangle, given base of operations and distance
• Isosceles triangle, given leg and apex angle
• Equilateral triangle
• thirty-lx-xc triangle, given the hypotenuse
• Triangle, given three sides (sss)
• Triangle, given one side and adjacent angles (asa)
• Triangle, given two angles and not-included side (aas)
• Triangle, given two sides and included angle (sas)
• Triangle medians
• Triangle midsegment
• Triangle distance
• Triangle altitude (outside case)

Right triangles

• Right Triangle, given 1 leg and hypotenuse (HL)
• Right Triangle, given both legs (LL)
• Right Triangle, given hypotenuse and one angle (HA)
• Right Triangle, given i leg and one angle (LA)

Triangle Centers

• Triangle incenter
• Triangle circumcenter
• Triangle orthocenter
• Triangle centroid

Circles, Arcs and Ellipses

• Finding the heart 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 1 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
• Detect the heart of a circle with any right-angled object

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