Lesson 5

Triangles in Circles

  • Let’s see how perpendicular bisectors relate to circumscribed circles.

5.1: One Perpendicular Bisector

The image shows a triangle.

Triangle A B C. Sides A C and A B are almost the same length and side B C is shorter. Side A B is horizontal and point C is above and to the right of middle of side A B.
  1. Construct the perpendicular bisector of segment \(AB\).
  2. Imagine a point \(D\) placed anywhere on the perpendicular bisector you constructed. How would the distance from \(D\) to \(A\) compare to the distance from \(D\) to \(B\)? Explain your reasoning.

5.2: Three Perpendicular Bisectors

  1. Construct the perpendicular bisector of segment \(BC\) from the earlier activity. Label the point where the 2 perpendicular bisectors intersect as \(P\).
  2. Use a colored pencil to draw segments \(PA,PB,\) and \(PC\). How do the lengths of these segments compare? Explain your reasoning.
  3. Imagine the perpendicular bisector of segment \(AC\). Will it pass through point \(P\)? Explain your reasoning.
  4. Construct the perpendicular bisector of segment \(AC\).
  5. Construct a circle centered at \(P\) with radius \(PA\).
  6. Why does the circle also pass through points \(B\) and \(C\)?


Points \(A,B,\) and \(C\) are graphed. Find the coordinates of the circumcenter and the radius of the circumscribed circle for triangle \(ABC\).

3 points on a grid, origin, O. X axis 0 to 12, by 2s. Y axis 0 to 12, by 2s. Point A 3 comma 3, point B 2 comma 10, point C 9 comma 11.

5.3: Wandering Centers

Move the vertices of triangle \(ABC\) and observe the resulting location of the triangle’s circumcenter, point \(D\). Determine what seems to be true when the circumcenter is in each of these locations:

 
  1. outside the triangle
  2. on one of the triangle’s sides
  3. inside the triangle

Summary

We saw that some quadrilaterals have circumscribed circles. Is the same true for triangles? In fact, all triangles have circumscribed circles. The key fact is that all points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment.

Suppose we have triangle \(ABC\) and we construct the perpendicular bisectors of all 3 sides. These perpendicular bisectors will all meet at a single point called the circumcenter of the triangle (label it \(D\)). This point is on the perpendicular bisector of \(AB\), so it’s equidistant from \(A\) and \(B\). It’s also on the perpendicular bisector of \(BC\), so it’s equidistant from \(B\) and \(C\). So, it is actually the same distance from \(A,B,\) and \(C\). We can draw a circle centered at \(D\) with radius \(AD\). The circle will pass through \(B\) and \(C\) too because the distances \(BD\) and \(CD\) are the same as the radius of the circle.

Circle with center D. Triangle ABC inside circle. Dashed lines from D to triangle sides create 90 degree angles. 

In this case, the circumcenter ​​​happened to fall inside triangle \(ABC\), but that does not need to happen. The images show cases where the circumcenter is inside a triangle, outside a triangle, and on one of the sides of a triangle.

Circle with center dot. Triangle is inside circle.
Circle with center point. Triangle inside circle but does not include point.
Circle with center point. Right riangle inside circle. Hypotenuse on center point.

Glossary Entries

  • circumcenter

    The circumcenter of a triangle is the intersection of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle.

  • circumscribed

    We say a polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.

  • cyclic quadrilateral

    A quadrilateral whose vertices all lie on the same circle.