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3.6 Triangles

Recall that Theorem 7, the Median Concurrence Theorem, established the centroid of the triangle. The centroid is the “center of mass” of a triangle. If the triangle and its interior were made of metal with perfectly constant mass, thickness and density, the centroid would be the balance point for that object. We have several triangle concurrence theorems and several notions of the center of a circle:

Table 1 Concurence Theorems Concepts

Concurrence

Point

Idea

Median

Centroid

Center of Mass

Perpendicular Bisector

Circumcenter

Center of circumscribing circle

Altitude

Orthocenter

Angle Bisector

Incenter

Center of incscribed circle
Circumscribe

A circle which contains all vertices of a polygon is said to \textbf{circumscribe} it.

Incircle

A circle which can be inscribed in a triangle (e.g. tangent to all three sides of the triangle and otherwise interior to the triangle).

Altitude

The perpendicular line sement from the vertex of an angle to the side opposite.

Circumcenter

The center of the unique circle that circumscribes a triangle.

Incenter

The center of the unique circle inscribed in a triangle.

Theorems

Proving Concurrence Theorems

Let two of the lines exist, then prove the third line passes through their intersection point. For the angle bisector concurrence, for example, we would start with an arbitrary \(\triangle ABC\) and construct the angle bisectors of \(\angle A\) and \(\angle B\). If these two lines are not parallel (you can prove they are not), then they intersect. Label this intersection \(P\) and prove that the angle bisector of \(\angle C\) passes through it.

  1. Perpendicular Bisector Concurrence. The three perpendicular bisectors of the sides a triangle are concurrent at the circumcenter of the triangle.

    Warning

    You must prove the intersection point of the perpendicular bisectors is also the center of the circumscribing circle. Prove the first result, then see if the relationships you need to prove the second are available.

    Note

    We often instruct students to think intuitively about the distance from a point to a line as the “minimum distance” from \(P\) to \(l\). This idea is equivalent to the definition because any other line segment drawn from \(P\) to \(l\) would be the hypotenuse of a right triangle. The Triangle Inequality then implies the new segment has a length greater than the distance from \(P\) to \(l\).

  2. A point is on the bisector of angle if and only if it is equidistant from both rays of the angle.

  3. Angle Bisector Concurrence. The bisectors of the interior angles of a triangle are concurrent at the incenter of the triangle.

    Warning

    You must prove the intersection point of the angle bisectors is also the center of the inscribed circle. Prove the first result, then see if the relationships you need to prove the second are available.

  4. Altitude Concurrence. The three altitudes of a triangle are concurrent at a point called the orthocenter.

  5. An angle bisector of an interior angle of a triangle is concurrent with the angle bisectors of the exterior angles of the two remaining angles of the triangles at a point called the excenter.

    Note

    This construction will lead to three distinct excenters.

  6. A triangle has three excenters, each of which is the center of a circle which is externally tangent to one side of the triangle and tangent to the extensions of the other two sides.

  7. In an equilateral triangle, the incenter, circumcenter, centroid and orthocenter are all concurrent.

  8. Euler Line. The orthocenter, centroid and circumcenter of a triangle are collinear. The line that contains these three points is called the Euler Line.

Note

The Euler Line actually contains several interesting points not mentioned in the above theorem, and investigating the relationships along this line (which are true for all triangles) would make an interesting geometry project.

3.5 Circles 3.7 Cyclic Quadrilaterals