To inscribe a circle, we have to find the triangle's incenter. We can do that by constructing angle bisectors to at least two of the triangle's vertexes. The point where the angle bisectors intersect is the triangle's incenter.
Finding the Incenter
Let's start by finding the angle bisector for ∠ T. Draw an arc across ST and TU with an arbitrary radius.
Using the intersections of the first arc with ST and UT as our centers, draw two smaller intersecting arcs. Make sure you keep the compass settings the same when you draw them.
The segment from T and through the intersection point of the two smaller arcs is the angle bisector to ∠ T.
We need one more angle bisector to find the incenter. Let's repeat the procedure for ∠ S.
Where the angle bisectors intersect, we find the triangle's incenter. Let's isolate this in our diagram.
To inscribe a circle, we also need to find the perpendicular line from the incenter and one of the triangle's sides.
Finding a Perpendicular Line
From the incenter, draw an arc that intersects TU, as well as two smaller arcs using the intersections of the first arc with TU as our centers. Make sure you keep the compass settings the same when drawing the two smaller arcs.
The segment from D to the intersection of the two smaller arcs is the perpendicular line from D to UT.
Constructing the Inscribed Circle
Finally, we will place the compass at D and set its width to DE. Now we can construct the inscribed circle.
Circumscribed Circle
To construct the circumscribed circle, we have to find the triangle's circumcenter. For this purpose, we have to construct perpendicular bisectors to at least two sides.
Finding Perpendicular Bisectors
Let's find the perpendicular bisector for ST. Open up a compass so that its width is greater than half the distance of ST, and draw two intersecting arcs using S and T as centers.
The line that contains both points of intersection is the perpendicular bisector to ST.
Let's repeat the process for US.
Where the perpendicular bisectors intersect, we find the triangle's circumcenter.
Constructing the Circumscribed Circle
The circumcenter is equidistant to all of the triangle's vertices. By setting the width of your compass to the distance between the circumcenter and an arbitrary vertex, we can subsequently draw a circle that passes through all three vertices.
