Gems of Geometry

The following series of video sessions are for students aiming for examinations like PRMO, RMO, INMO and IMO. Students and other Maths enthusiasts may also find this series useful to satiate their intellectual curiosity. This series comprises of theorems and results in High School Geometry which are usually not included in regular Maths curricula.

Concept Lecture

Session # 01 : The Laws of Sine

Session # 02 : Ceva's Theorem

Worksheet # 01 : Worksheet on Sine Law and Ceva's Law

Session # 03 : Stewart's Theorem

Session # 04 : Medians of a triangle divide it into six parts of equal areas.

Session # 05 : Medians of a triangle trisect each other

Session # 06 : Each angle bisector of a triangle divides the opposite side into segments proportional in length to the adjacent sides. Proof using Sine Rule

Worksheet # 02 : Worksheet on Stewart's Theorem and properties of medians of a triangle

Session # 07 : The area of a triangle is equal to the product of the semi-perimeter and the in-radius

Session # 08 : The locus of a point equidistant from two intersecting lines is the angle bisector of the angle between the intersecting lines.

Session # 09 : Theorem: The external bisectors of any two angles of a triangle are concurrent with the internal bisector of the third angle

Session # 10 : Some properties of ex-circles and in-circle of a triangle

Session # 11 : If two chords of a circle subtend different acute angles at points on the circle, the smaller angle belongs to the shorter chord.

Session # 12 : In a triangle, the angle bisector of the smaller angle is greater than the angle bisector of the greater angle

Session # 13 : The Steiner-Lehmus Theorem : Any triangle that has two equal angle bisectors (each measured from a vertex to the opposite side) is isosceles.

Session # 14 : The orthocenter of an acute-angled triangle is the in-center of its orthic triangle.

Session # 15 : What is a medial triangle? What is an Euler Line in a triangle?

Session # 16 : Centroids of a triangle and its medial triangle are coincident

Session # 17 : The Circumcenter of a Triangle and the Orthocenter of its Medial Triangle are coincident

Session # 18 : The orthocenter, centroid and circumcenter of any triangle are collinear. The centroid divides the distance from the orthocenter to the circumcenter in the ratio 2:1

Session # 19 : Construction of a Nine-Point Circle using GeoGebra

Session # 20 : Theorem: The feet of the three altitudes of any triangle, the mid-points of the three sides, and the mid-points of the segments from the three vertices to the orthocenter, all lie on the same circle, of radius 1/2 R