How to Draw a Circle in a Triangle

This page 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 use every other vertex instead of all six.

Every bit can exist seen in Definition of a Hexagon, each side of a regular hexagon is equal to the altitude from the center to whatsoever vertex. This construction simply sets the compass width to that radius, and so steps that length off effectually the circle to create the six vertices of a hexagon.

But instead of drawing a hexagon, we employ every other vertex to make a triangle instead. Since the hexagon construction effectively divided the circle into 6 equal arcs, past using every other signal, we divide it into 3 equal arcs instead. The three chords of these arcs course the desired equilateral triangle.

Another style of thinking most information technology 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 terminal cartoon from the above animation, only with actress lines and the vertices labelled.

	Statement	Reason
NOTE: Steps 1 through 7 are the same as for the construction of a hexagon inscribed in a circle. In the case of an inscribed equilateral triangle, we use every other point on the circumvolve.
one	A,B,C,D,E,F all lie on the circumvolve center O	Past structure.
2	AB = BC = CD = DE = EF	They were all fatigued with the same compass width.
From (ii) we come across that five sides are equal in length, but the last side FA was not drawn with the compasses. It was the "left over" space every bit we stepped around the circle and stopped at F. So we have to testify it is congruent 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).
4	yard∠AOB = lx°	All interior angles of an equilateral triangle are 60°.
5	m∠AOF = 60°	As in (4) m∠BOC, m∠COD, m∠DOE, m∠EOF are all &60deg; Since all the central angles add to 360°, m∠AOF = 360 - five(sixty)
6	Triangle BOA, AOF are congruent	SAS Run across Test for congruence, side-bending-side.
7	AF = AB	CPCTC - Corresponding Parts of Coinciding Triangles are Congruent
So at present we can prove that BDF is an equilateral triangle
eight	All six fundamental angles (∠AOB, ∠BOC, ∠COD, ∠DOE, ∠EOF, ∠FOA) are congruent	From (iv) and by repetition for the other v angles, all half dozen angles have a mensurate of 60°
ix	The angles ∠BOD, ∠DOF, ∠BOF are coinciding	From (8) - They are each the sum of 2 60° angles
10	Triangles BOD, DOF and BOF are congruent.	The sides are all equal radii of the circumvolve, and from (9), the included angles are congruent. Come across Examination for congruence, side-angle-side
11	BDF is an equilateral triangle.	From (10) 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 circle	From (11) and all three vertices B,D,F lie on the given circle.

- Q.Due east.D

Endeavour it yourself

Click here for a printable worksheet containing two problems to endeavor. When you lot go to the folio, use the browser print command to print every bit 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 due north line segments
• Deviation of 2 line segments
• Perpendicular bisector of a line segment
• Perpendicular at a point on a line
• Perpendicular from a line through a betoken
• Perpendicular from endpoint of a ray
• Divide a segment into due north equal parts
• Parallel line through a indicate (angle copy)
• Parallel line through a point (rhomb)
• Parallel line through a point (translation)

Angles

• Bisecting an angle
• Re-create an bending
• Construct a 30° angle
• Construct a 45° bending
• Construct a lx° angle
• Construct a 90° angle (correct bending)
• Sum of n angles
• Difference of 2 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 of operations 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 ane side and side by side 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)
• Correct 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 ii circles (internal)
• Incircle of a triangle
• Focus points of a given ellipse
• Circumcircle of a triangle

Polygons

• Square given ane side
• Square inscribed in a circle
• Hexagon given 1 side
• Hexagon inscribed in a given circumvolve
• Pentagon inscribed in a given circle

Non-Euclidean constructions

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

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