Inscribed & Circumscribed
Circles of a Triangle
Master the construction of incircles and circumcircles, understand their geometric properties, and prove angle relationships for inscribed quadrilaterals.
Geometry
What Are Inscribed & Circumscribed Circles?
Inscribed Circle (Incircle)
The largest circle that fits inside a triangle, tangent to all three sides. Its center is called the incenter.
Circumscribed Circle (Circumcircle)
The unique circle that passes through all three vertices of a triangle. Its center is called the circumcenter.
The Incenter: Where Angle Bisectors Meet
The Circumcenter: Where Perpendicular Bisectors Meet
Core Theorems for Construction
1
Angle Bisector Concurrency: The three angle bisectors of any triangle are concurrent. Their common point is the incenter, equidistant from all three sides.
2
Perpendicular Bisector Concurrency: The three perpendicular bisectors of any triangle are concurrent. Their common point is the circumcenter, equidistant from all three vertices.
3
Inscribed Angle Theorem: An inscribed angle is half the central angle that subtends the same arc: \(\theta_{\text{inscribed}} = \frac{1}{2}\theta_{\text{central}}\).
Quadrilateral Inscribed in a Circle
Incircle vs. Circumcircle
| Property | Incircle | Circumcircle |
|---|---|---|
| Center | Incenter (\(I\)) | Circumcenter (\(O\)) |
| Found by intersecting | Angle bisectors | Perpendicular bisectors |
| Radius measures distance to | Sides (tangent) | Vertices (passes through) |
| Position relative to triangle | Always inside | Inside (acute), on hypotenuse (right), outside (obtuse) |
| Radius formula | \(r = \frac{\text{Area}}{s}\) | \(R = \frac{abc}{4 \cdot \text{Area}}\) |
Where Does the Circumcenter Fall?
Acute Triangle
Circumcenter lies inside the triangle. All angles \(< 90^\circ\).
Right Triangle
Circumcenter lies on the midpoint of the hypotenuse. One angle \(= 90^\circ\).
Obtuse Triangle
Circumcenter lies outside the triangle. One angle \(> 90^\circ\).
How to Construct the Incircle
1
Bisect two angles. Using a compass, construct the angle bisector of at least two angles of the triangle.
2
Mark the incenter. The point where the bisectors intersect is the incenter \(I\).
3
Drop a perpendicular. From \(I\), construct a perpendicular to any side of the triangle. The foot of the perpendicular is the tangent point.
4
Draw the incircle. Set your compass to the distance from \(I\) to that foot, then draw the circle centered at \(I\).
How to Construct the Circumcircle
1
Find two perpendicular bisectors. Construct the perpendicular bisector of at least two sides of the triangle.
2
Mark the circumcenter. The point where the perpendicular bisectors intersect is the circumcenter \(O\).
3
Measure the circumradius. Set your compass from \(O\) to any vertex (they are all equidistant).
4
Draw the circumcircle. Draw the circle centered at \(O\) passing through all three vertices.
Example 1: Finding the Inradius
Example 2: Finding the Circumradius
Example 3: Inscribed Quadrilateral Angles
Why Opposite Angles Sum to \(180^\circ\)
1
Inscribed Angle Theorem: An inscribed angle equals half its intercepted arc.
2
\(\angle A\) intercepts \(\text{arc } BCD\), and \(\angle C\) intercepts \(\text{arc } BAD\).
3
Together, \(\text{arc } BCD\) and \(\text{arc } BAD\) form the full circle: \(\text{arc } BCD + \text{arc } BAD = 360^\circ\).
4
Therefore \(\angle A + \angle C = \frac{1}{2}(\text{arc } BCD) + \frac{1}{2}(\text{arc } BAD) = \frac{1}{2}(360^\circ) = 180^\circ\). ✓
Common Mistakes to Avoid
✕ Mixing up bisectors
Angle bisectors give the incenter. Perpendicular bisectors give the circumcenter. Swapping them yields the wrong center and wrong circle.
✕ Adjacent instead of opposite
The supplementary angle theorem applies to opposite angles of an inscribed quadrilateral, not adjacent ones.
✕ Circumcenter always inside
The circumcenter is outside the triangle when the triangle is obtuse. Only the incenter is always interior.
Summary: Key Facts at a Glance
| Concept | Key Detail |
|---|---|
| Incircle center | Intersection of angle bisectors |
| Circumcircle center | Intersection of perpendicular bisectors |
| Inradius formula | \(r = \frac{\text{Area}}{s}\) |
| Circumradius formula | \(R = \frac{abc}{4 \cdot \text{Area}}\) |
| Inscribed quadrilateral | Opposite angles sum to \(180^\circ\) |
| Inscribed angle | Half the central angle on the same arc |
Question 1 of 5
The incenter of a triangle is found by constructing the intersection of which lines?
The incenter is the point where the three angle bisectors of a triangle meet. Perpendicular bisectors give the circumcenter, medians give the centroid, and altitudes give the orthocenter.
Question 2 of 5
A triangle has sides \(a = 5\), \(b = 12\), \(c = 13\) and area \(= 30\). What is the inradius \(r\)?
The semi-perimeter is \(s = \frac{5+12+13}{2} = 15\). Then \(r = \frac{\text{Area}}{s} = \frac{30}{15} = 2\).
Question 3 of 5
Quadrilateral PQRS is inscribed in a circle. If \(\angle P = 110^\circ\), what is \(\angle R\)?
Opposite angles in an inscribed quadrilateral are supplementary: \(\angle P + \angle R = 180^\circ\), so \(\angle R = 180^\circ - 110^\circ = 70^\circ\).
Question 4 of 5
In an obtuse triangle, where does the circumcenter lie?
When a triangle is obtuse, the circumcenter falls outside the triangle, on the opposite side of the longest edge from the obtuse vertex. It only lies on a side (the hypotenuse midpoint) for a right triangle.
Question 5 of 5
A triangle has sides \(a = 7\), \(b = 24\), \(c = 25\) and area \(= 84\). What is the circumradius \(R\)?
\(R = \frac{abc}{4 \cdot \text{Area}} = \frac{7 \cdot 24 \cdot 25}{4 \cdot 84} = \frac{4200}{336} = 12.5\). Note: this is a right triangle (\(7^2 + 24^2 = 25^2\)), so the circumradius is half the hypotenuse: \(\frac{25}{2} = 12.5\).