# 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.

## Section 1

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

Session # 21 : The radius of the Nine-Point Circle is 1/2 R

Session # 22 : Construction of a **Pedal Triangle**.

Session # 23 : If x, y and z are the distances of a pedal point P from the three vertices A, B and C of a ∆ ABC, then the pedal triangle has sides ax/2R, by/2R & cz/2R

Session # 24 : A Triangle and its 3rd Pedal Triangle are similar.

## Section 2

Session # 25 : If two chords BE and CD of a circle intersect at F then BF x FE = CF x FD.

Session # 26 : PT^2 = PA x PA' where P is a point outside a circle, PT is a tangent and PAA' is a secant of the circle.

Session # 27 : Power of a point w.r.t a circle is the difference of the square of its distance from the centre (d) and the square of the radius of the circle (R). d^2 = R^2 - 2Rr; where r is the in-radius

Session # 28 : **Power of a Point** with respect to a Circle

Session # 29 : **Coaxal Circles**

Session # 30 : The altitude of a triangle is equal to product of the adjacent sides divided by twice the circum-radius.

Session # 31 : **Radical Axis** of circles drawn on two cevians of a triangle as diameters passes through the orthocenter.

Session # 32 : **Simson's Line** is the line joining the feet of the perpendiculars dropped from a point on the circumcircle of a triangle on the sides of the triangle.

Session # 33 : The feet of the perpendiculars from a point to the sides of a triangle are collinear if and only if the point lies on the circumcircle.