A Mathematical Blog

There are quite a few circles associated with any given triangle. A circumcircle for example, is the name of the circle that passes through the three vertices of a triangle.  There is also an incircle which is the circle of the largest possible radius that can fit inside a triangle. Another one is the nine-point circle. It is the circle that passes through the nine points in the plane of a triangle, hence the name. These nine points are :  Feet of the three altitudes(an altitude is a segment from a vertex perpendicular to a line containing the side opposite to the vertex) Midpoints of the three sides of triangle(also said to be the points where the three medians meet the sides) Midpoints of the three line segments from the vertices to the orthocenter - the point where the three altitudes of a triangle intersect.  In other words, these nine points are concyclic for any given triangle. Meaning, for a given triangle they lie on the circumference of the circle - the nin...

Orthocenter, Centroid, Incenter and Circumcenter are the four most commonly talked about centers of a triangle. I have written a great deal about the Incenter, the Circumcenter and the Centroid in my previous posts. In this post, I will be specifically writing about the Orthocenter.  Orthocenter is the point in the plane of a triangle where all three altitudes of the triangle intersect. An altitude of a triangle is the segment from a vertex perpendicular to the side opposite to the vertex, or perpendicular to the line that contains the opposite side. In the figure below, AD is the altitude from vertex A perpendicular to BC(or the line containing the side BC), the side sitting opposite of vertex A, of △ABC.  Similarly, BE and DF are the other two altitudes of triangle ABC emanating from vertices B and C. All three altitudes intersect at point H - the Orthocenter of the triangle.  In any triangle, the three altitudes are always concurrent(intersecting at a singl...

Ceva’s Theorem is a fundamental theorem related to triangles. It was proved by an Italian mathematician Giovanni Ceva. Even though he may not have been the first to prove it, the theorem still bears his name.  I will write about one of many (pdf file) proofs of Ceva’s theorem and its converse in some other related post. This post is all about understanding the patterns in the mathematical relation given by the theorem . A cevian is a line segment joining a vertex of a triangle to any point on the line which contains the opposite side of the vertex. A cevian can lie outside of a triangle too, which we generally find in the case of obtuse angled triangles. ( NOTE : Is the line segment overlapping an edge of a triangle a cevian? I have attempted to answer this question at the end of this post.) Sometimes a cevian is also defined as a line(instead of a segment) that passes through a vertex and cuts the line which contains the side opposite to the vertex.  In △ABC in the...