To inscribe a circle, you need to find the triangle's incenter. To circumscribe a circle, you need to find the triangle's circumcenter.
Inscribed Circle:
Circumscribed Circle:
Practice makes perfect
Below, we have drawn an arbitrary right triangle.
Method: See solution.
Inscribed Circle
To inscribe a circle, we have to find the triangle's incenter. We can do that by drawing angle bisectors for at least two of the triangle's vertices. The point where the angle bisectors intersect is the triangle's incenter.
Finding the Incenter
Let's start by drawing an angle bisector to ∠ E. First, we will draw an arc across DE and FE with an arbitrary radius like below.
Next, we draw two smaller arcs using the intersections of the first arc with FE and DE as our midpoints. Make sure you keep the compass settings the same for both arcs.
The segment from E and through the intersection point of the two smaller arcs is the angle bisector to ∠ E.
We need one more angle bisector to find the incenter. Let's repeat the procedure for ∠ F.
We find the triangle's incenter where the angle bisectors intersect. 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
To find the perpendicular line to, for example, FE, we have to draw an arc that intersects FE at two points. Using the intersections of the first arc with FE as centers, we draw a pair of intersecting arcs using the same compass setting.
The segment from G to the intersection of the two smaller arcs is the perpendicular line to FE.
Constructing the Inscribed Circle
Finally, we will place the compass at G and set its width to GH. Now we can construct the inscribed circle.
Let's find the perpendicular bisector for DE. Open a compass until its width is greater than half the distance of DE, and draw two intersecting arcs using D and E as centers.
The line that contains both points of intersection is the perpendicular bisector to DE.
Let's repeat the process for FE.
Where the perpendicular bisectors intersect, we find the triangle's circumcenter.
Constructing the Circumscribed Circle
The circumcenter is equidistant to the triangle's vertices. Therefore, by setting our compass width to the distance between the circumcenter and an arbitrary vertex, we can draw a circle that passes through all three vertices.
