Constructing Inscribed and Circumscribed Circles - Geometry

1. If an incircle touches side $\overline{AB}$ at $T$, what is the angle between $IT$ and $\overline{AB}$?

   $90^\circ$. Radius to tangency point is perpendicular to the side.

2. If $ABCD$ is cyclic and $m\angle D = 103^\circ$, what is $m\angle B$?

   $77^\circ$. Opposite angles sum to $180^\circ$: $180^\circ - 103^\circ = 77^\circ$.

3. What is the circumradius of a triangle?

   The distance from the circumcenter to any vertex. All vertices are equidistant from the circumcenter.

4. What is a cyclic quadrilateral?

   A quadrilateral with all four vertices on one circle. Also called an inscribed quadrilateral.

5. What is the key angle fact used to prove opposite angles in a cyclic quadrilateral are supplementary?

   Inscribed angles equal half their intercepted arcs. Since opposite arcs sum to $360^\circ$, opposite angles sum to $180^\circ$.

6. What is the incircle of a triangle?

   The circle tangent to all three sides of the triangle. The incenter is the center with inradius as radius.

7. State the opposite-angle property of a cyclic quadrilateral $ABCD$.

   $m\angle A + m\angle C = 180^\circ$ and $m\angle B + m\angle D = 180^\circ$. Opposite angles are supplementary in cyclic quadrilaterals.

8. If $ABCD$ is cyclic and the exterior angle at $A$ is $61^\circ$, what is $m\angle C$?

   $61^\circ$. The exterior angle equals the remote interior angle.

9. What point is equidistant from all three vertices of a triangle?

   The circumcenter. The circumcenter is where perpendicular bisectors meet.

10. What is the relationship between a radius and a tangent line at the point of tangency?

    The radius is perpendicular to the tangent at the point of tangency. Radius and tangent form right angles at tangency.

11. What is the angle-bisector theorem statement for point $D$ on $\overline{BC}$ with $AD$ bisecting $\angle A$?

    $\frac{BD}{DC} = \frac{AB}{AC}$. The angle bisector divides the opposite side proportionally.

12. What is the measure of an inscribed angle intercepting a $100^\circ$ arc?

    $50^\circ$. Apply the inscribed angle theorem: $\frac{100^\circ}{2} = 50^\circ$.

13. What is the defining property of points on the angle bisector of $\angle A$?

    They are equidistant from the sides of $\angle A$. This is the angle bisector theorem property.

14. In a right triangle, what is the circumradius in terms of hypotenuse length $c$?

    $R = \frac{c}{2}$. Circumcenter lies at hypotenuse midpoint in right triangles.

15. What is the definition of a perpendicular distance from a point to a line?

    The length of the segment from the point to the line forming a $90^\circ$ angle. This gives the shortest distance from point to line.

16. Which lines must be constructed to get the center of the incircle of $\triangle ABC$?

    Two angle bisectors of the triangle. Their intersection locates the incenter.

17. Where is the incenter located in any nondegenerate triangle?

    Inside the triangle. The incenter always lies within the triangle's interior.

18. Identify the circle used to circumscribe $\triangle ABC$ once the circumcenter $O$ is found.

    Circle centered at $O$ with radius $OA$. All vertices must lie on this circle.

19. What point is equidistant from all three sides of a triangle?

    The incenter. The incenter is where angle bisectors meet.

20. Identify the circle used to inscribe in $\triangle ABC$ once the incenter $I$ is found.

    Circle centered at $I$ with radius equal to the distance to a side. The circle must be tangent to all three sides.

21. Which construction lines locate a triangle's circumcenter?

    Perpendicular bisectors of at least two sides. Two lines are sufficient since they intersect at one point.

22. What is the measure of an angle subtending a diameter in a circle?

    $90^\circ$. Angles inscribed in a semicircle are right angles.

23. What is the relationship between a central angle and its intercepted arc?

    Central angle measure equals intercepted arc measure. Central angles directly measure their intercepted arcs.

24. In a right triangle with hypotenuse $c = 10$, what is the circumradius $R$?

    $R = 5$. Apply the formula $R = \frac{c}{2} = \frac{10}{2} = 5$.

25. If $AD$ bisects $\angle A$ and $AB = 6$, $AC = 9$, what is $\frac{BD}{DC}$?

    $\frac{BD}{DC} = \frac{2}{3}$. Apply angle bisector theorem: $\frac{BD}{DC} = \frac{AB}{AC} = \frac{6}{9} = \frac{2}{3}$.

26. What is the orthocenter of a triangle?

    The intersection of the three altitudes of the triangle. This point may lie inside, outside, or on the triangle.

27. Which lines must be constructed to get the center of the circumcircle of $\triangle ABC$?

    Two perpendicular bisectors of the sides. Their intersection locates the circumcenter.

28. What is an angle bisector?

    A ray that divides an angle into two congruent angles. This creates two angles of equal measure.

29. If $ABCD$ is cyclic and $m\angle A = 68^\circ$, what is $m\angle C$?

    $112^\circ$. Opposite angles sum to $180^\circ$: $180^\circ - 68^\circ = 112^\circ$.

30. What is the inradius of a triangle?

    The perpendicular distance from the incenter to any side. All sides are equidistant from the incenter.

31. What is the definition of a circumscribed circle about a polygon?

    A circle passing through all vertices of the polygon. Also called the circumcircle of the polygon.

32. What is the circumcenter of a triangle?

    The intersection of the perpendicular bisectors of the sides. This point is equidistant from all vertices.

33. If $AC$ is a diameter and $B$ is on the circle, what is $m\angle ABC$?

    $90^\circ$. Thales' theorem: angles in semicircle are $90^\circ$.

34. Identify whether $ABCD$ is cyclic if $m\angle B = 95^\circ$ and $m\angle D = 70^\circ$.

    No, because $95^\circ + 70^\circ \ne 180^\circ$. The sum is $165^\circ$, not $180^\circ$, so not cyclic.

35. Identify whether $ABCD$ is cyclic if $m\angle A = 92^\circ$ and $m\angle C = 88^\circ$.

    Yes, because $92^\circ + 88^\circ = 180^\circ$. Opposite angles are supplementary, confirming it's cyclic.

36. Where is the circumcenter located in an acute triangle?

    Inside the triangle. All angles are less than $90^\circ$ in acute triangles.

37. A tangent-chord angle intercepts a $140^\circ$ arc. What is the angle measure?

    $70^\circ$. Apply tangent-chord theorem: $\frac{140^\circ}{2} = 70^\circ$.

38. What is the definition of an inscribed circle in a polygon?

    A circle tangent to each side of the polygon. Also called the incircle of the polygon.

39. What is the centroid of a triangle?

    The intersection of the three medians of the triangle. This point divides each median in a $2:1$ ratio.

40. What is the incenter of a triangle?

    The intersection of the angle bisectors of the triangle. This point is equidistant from all sides.

41. Where is the circumcenter located in an obtuse triangle?

    Outside the triangle. One angle exceeds $90^\circ$ in obtuse triangles.

42. What is a perpendicular bisector of a segment?

    A line perpendicular to the segment at its midpoint. This creates two congruent right triangles.

43. What is the measure of a central angle intercepting a $120^\circ$ arc?

    $120^\circ$. Central angle equals its intercepted arc measure.

44. What is the circumcircle of a triangle?

    The circle through all three vertices of the triangle. The circumcenter is the center with circumradius as radius.

45. What is the tangent-chord angle theorem (angle formed by tangent and chord)?

    The angle equals half the measure of its intercepted arc. This applies to angles formed by tangent and chord.

46. Which construction lines locate a triangle's incenter?

    Angle bisectors of at least two angles. Two lines are sufficient since they intersect at one point.

47. What is the defining property of points on a perpendicular bisector of $\overline{AB}$?

    They are equidistant from $A$ and $B$. This is the perpendicular bisector theorem.

48. What is the arc measure intercepted by an inscribed angle of $35^\circ$?

    $70^\circ$. Apply the inscribed angle theorem: $35^\circ \times 2 = 70^\circ$.

49. Where is the circumcenter located in a right triangle?

    At the midpoint of the hypotenuse. The hypotenuse is a diameter of the circumcircle.

50. If $ABCD$ is cyclic, what is the relationship between exterior angle at $A$ and interior opposite angle $C$?

    An exterior angle equals the interior opposite angle: $m\angle A_{ext} = m\angle C$. Exterior and remote interior angles are equal in cyclic quadrilaterals.

51. What condition on angles proves a quadrilateral is cyclic?

    A pair of opposite angles is supplementary. This is the converse of the cyclic quadrilateral theorem.

52. What is the relationship between an inscribed angle and its intercepted arc?

    Inscribed angle measure is half the intercepted arc measure. This is the inscribed angle theorem.

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