Concept of Incenter,Incircle,Median,Centroid their proof and Question Solving

In this Video lecture I have explained in detail that how a circle in formed inside a triangle with its center being the intersection point of angular bisectors drawn from the three vertices. Till the last video I told five formulas related to the triangles. In this video I have proved the sixth important formula which is related to the study of triangles. The formula is -

Area of the Triangle = rs

where r = Inradius of the triangle and s = semiperimeter of the triangle.

After that I have proved one more formula on whose basis questions have been asked numerous times in competitive examination. It is - 

Angle AIC = 90 + Angle B/2

After that I have explained that why a median divides a triangle in two triangles of equal area and why three medians divide the whole triangle into six smaller triangles of equal area. Also the proof of why the Centroid divides a median in the ratio of 2:1 has been proved in this video. After that the below questions have been explained in the video. Some questions around four I was not able to cover in this video due to limited file size so that I would take up in my next video on geometry.

Question 1. X and Y are points on the side AB & AC of a triangle ABC. If AX = 2 cm. XB = 4 cm. AY = 3 cm. YC = 6 cm. Find the ratio of side BC and XY.

Question 2. If the triangle ABC is right angled at B, find its circumradius if the sides AB and BC are 15 cm and 20 cm respectively.

Question 3. Find the ratio of the area of equilateral triangle drawn with the side of a square as its base and the area of the equilateral triangle described on the diagonal of the square.

Question 4. The perimeters of two similar triangles are 30 cm and 20 cm respectively. If the side of the first triangle is 12 cm then find out the proportional side of the other triangle.

Question 5. In a triangle XYZ, XY² + XZ² = 200 square cm. The length of the median AX is 10 cm. Find the length of YZ.