The triangle is one of the most frequently discussed geometric figures in the study of geometry. Its relationship with circles is fundamental to understanding various geometry results. Some of these results are called notable points of the triangle.

Similarity of Triangles

In the proportions post, we saw concepts of similarity of triangles. Here we will show that the similarity of triangles can be demonstrated by circles.

Triangles inside the circle

Lemma 1: Let AB and CD be different chords of a circle that intersect at a point P inside the circle. Then

$$\dfrac{AP}{CP} = \dfrac{PD}{PB} $$

Triangles with two tangent edges

Lemma 2: If the two sides of a vertex angle P are tangent to a circle with center O at points A and B, then $PA = PB.$

Triangles with a tangent and a secant edge

Lemma 3: Let P be a point on the exterior of a circle C, r be a straight line through P secant to C at points B and C and s be a tangent line to C at A. Then

$$\dfrac{PB}{PA}=\dfrac{PA}{PC}$$

Triangles with two secant edges

Lemma 4: Let P be a point on the exterior of a circle C, r be a straight line through P secant to C at points A and B and s be a straight line secant to C at points C and D. Then

$$\dfrac{PA}{PC}=\dfrac{PD}{PB}$$

Notable Points in a Triangle

The properties of the notable points of triangles are often related to the circumference. Before presenting these points, let’s define some fundamental terms for geometry,

circumcenter

Definition (perpendicular bisector): It is the geometric locus of points that are equidistant from two distinct points A and B.

Theorem 1: The three perpendicular bisectors sides of any triangle meet at a point that is equidistant (same distance) from its vertices, and thus this point is the center of a circle that contains the three vertices.

Definition (circumcenter): It is the meeting point between your perpendicular bisectors.

incenter

Theorem 2: The angle bisectors of any triangle meet at a point in its interior equidistant from the sides. The circle centered at this point and whose radius is this common distance is tangent to the sides.

Definition (incenter): It is the meeting point between your angle bisectors.

orthocenter

Theorem 3: The (straight) heights of any triangle meet at a point.

Definition (orthocenter): It is the meeting point of the heights.

centroid

Theorem 4: The medians of any triangle meet at a point whose distance to any vertex is equal to 2/3 of the corresponding median measurement.

Definition (centroid): Given a triangle, the meeting point of its medians is called the centroid or centroid.

Euler straight line

Theorem 5 (Euler’s straight line): In any triangle, circumcenter, orthocenter and centroid are collinear.

These are the points on triangles and circles. Wait for the next posts.