Question

20 pts !!! Please give a detailed answer thanks :)

How do you construct the inscribed and circumscribed circles of a triangle?

Answer 1

• inscribed: centered at the intersection of angle bisectors, radius = distance to one side
• circumscribed: centered at the intersection of perpendicular bisectors, radius = distance to one vertex

Explanation:

The steps to constructing either circle start with finding the relevant center and radius. The required construction techniques are ...

• bisect an angle
• perpendicular to a line through an external point
• perpendicular bisector

Inscribed circle

Summary

The incenter (center of the inscribed circle) is located at the coincidence of the angle bisectors. The radius is the perpendicular distance to any side, so will lie on the line through the incenter and perpendicular to one side.

Angle bisectors

Reference the first attachment. We have elected to bisect the largest two angles of triangle ABC. In each case, we used these steps:

1. Centered at a vertex (B for example), draw an arc through the two sides of that angle (arc HI for example).
2. Centered at the points of intersection with the sides, draw two intersecting arcs with the same radius (arcs HQ and IQ).
3. Draw the angle bisector through this point of intersection and the original vertex (green line BQ).

After this process is repeated for another angle, the point of intersection of the two angle bisectors is the incenter (R).

Radius

Finding the radius requires finding the perpendicular distance to one side of the triangle. That is done by constructing a perpendicular to the side through the incenter.

1. Centered at the incenter (R), draw an arc that intersects one side in two places (arc UV).
2. Centered at the points of intersection with the side, draw two intersecting arcs with the same radius (arcs UZ and VZ. Z is the unlabeled point at upper right).
3. Draw perpendicular RZ intersecting the midpoint of UV at X.

Incircle

The inscribed circle is centered at R and has radius RX.

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Circumscribed Circle

Summary

The circumcenter (center of the circumscribed circle) is located at the coincidence of the bisectors of the sides of the triangle. The radius is the distance from the circumcenter to any vertex.

Perpendicular bisector

Reference the second attachment. We have elected to bisect the longest and shortest of the triangle's sides. The purpose of this choice is to make the perpendicular bisectors intersect at a large angle, so the point is as accurately located as possible.

1. Choose one side of the triangle and set the radius to more than half its length.
2. Using that radius, draw intersecting arcs using each end of the side as a center (side AB and arcs FG, for example).
3. Draw the perpendicular bisector through the points where the arcs intersect (line FG).

Repeat this process for another side (BC to create bisector JK). The intersection of the two bisectors (L) is the circumcenter.

Circumcircle

The circumscribed circle is centered at L and has radius LA.

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Additional comments

In general, arcs that are required to intersect each other will be drawn with the same radius. An arc that is required to intersect two lines, or one line in two places, may have any convenient radius.

As with any geometric construction, the tools required are a marking tool, a compass, and a straightedge. Usually, the preferred marking tool is a sharp pencil.

If the triangle is obtuse, the circumcenter will be outside the triangle (adjacent to the long side). If the triangle is a right triangle, the circumcenter is the midpoint of the hypotenuse.