How to Draw a Circle Inside a Equilateral Triangle

This page shows how to construct (draw) an equilateral triangle inscribed in a circumvolve with a compass and straightedge or ruler. This is the largest equilateral triangle that volition 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.

Equally tin can be seen in Definition of a Hexagon, each side of a regular hexagon is equal to the distance from the center to whatsoever vertex. This construction simply sets the compass width to that radius, and then steps that length off around the circumvolve to create the six vertices of a hexagon.

Simply instead of drawing a hexagon, we use every other vertex to make a triangle instead. Since the hexagon structure finer divided the circle into six equal arcs, by using every other point, nosotros split up it into three equal arcs instead. The three chords of these arcs grade the desired equilateral triangle.

Another manner of thinking about 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 final drawing from the in a higher place animation, only with actress lines and the vertices labelled.

	Argument	Reason
Notation: Steps 1 through 7 are the same as for the construction of a hexagon inscribed in a circle. In the example of an inscribed equilateral triangle, we use every other signal on the circle.
i	A,B,C,D,E,F all prevarication on the circumvolve centre O	By structure.
two	AB = BC = CD = DE = EF	They were all fatigued with the same compass width.
From (2) 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 equally we stepped around the circle and stopped at F. And so we accept to prove information technology 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	m∠AOB = 60°	All interior angles of an equilateral triangle are 60°.
5	m∠AOF = 60°	Equally in (4) m∠BOC, chiliad∠COD, m∠DOE, thousand∠EOF are all &60deg; Since all the central angles add to 360°, thousand∠AOF = 360 - 5(lx)
half-dozen	Triangle BOA, AOF are coinciding	SAS See Examination for congruence, side-angle-side.
vii	AF = AB	CPCTC - Corresponding Parts of Congruent Triangles are Congruent
So now we tin show that BDF is an equilateral triangle
viii	All six cardinal angles (∠AOB, ∠BOC, ∠COD, ∠DOE, ∠EOF, ∠FOA) are congruent	From (iv) and by repetition for the other 5 angles, all six angles accept a measure of 60°
9	The angles ∠BOD, ∠DOF, ∠BOF are congruent	From (8) - They are each the sum of two 60° angles
ten	Triangles BOD, DOF and BOF are congruent.	The sides are all equal radii of the circle, and from (ix), the included angles are congruent. See Test for congruence, side-angle-side
11	BDF is an equilateral triangle.	From (10) BD, DF, FB a re coinciding. CPCTC - Corresponding 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 (xi) and all three vertices B,D,F lie on the given circle.

- Q.Eastward.D

Endeavor information technology yourself

Click hither for a printable worksheet containing two problems to try. When you become to the page, use the browser print command to print as many as y'all wish. The printed output is non copyright.

Other constructions pages on this site

• List of printable constructions worksheets

Lines

• Introduction to constructions
• Copy a line segment
• Sum of northward 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 betoken
• Perpendicular from endpoint of a ray
• Divide a segment into n equal parts
• Parallel line through a point (angle copy)
• Parallel line through a betoken (rhombus)
• Parallel line through a point (translation)

Angles

• Bisecting an angle
• Re-create an angle
• Construct a 30° angle
• Construct a 45° angle
• Construct a sixty° angle
• Construct a ninety° angle (correct angle)
• 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 and distance
• Isosceles triangle, given leg and noon bending
• Equilateral triangle
• 30-sixty-ninety 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 example)

Correct triangles

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

Triangle Centers

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

Circles, Arcs and Ellipses

• Finding the eye of a circle
• Circle given 3 points
• Tangent at a point on the circle
• Tangents through an external point
• Tangents to 2 circles (external)
• Tangents to 2 circles (internal)
• Incircle of a triangle
• Focus points of a given ellipse
• Circumcircle of a triangle

Polygons

• Square given i side
• Foursquare 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 whatsoever right-angled object

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