Let O be a point inside a triangle ABC such that `/_OAB = /_OBC = /_OCA = omega,` then Show that:

If lengths of the sides of a triangle ABC are 4cm,5cm and 6cm,O is point inside the triangle ABC such that /_OBC=/_OCA=/_OAB=theta then value of cot theta is

O is any point inside a triangle ABC. The bisector of /_AOB,/_BOC and /_COA meet the sides AB,BC and CA in point D, and F respectively.Show that ADxBExCF =DBxECFA

Let P be a point inside a triangle ABC with /_ABC=90^(@) . Let P_(1) and P_(2) be the images of P under reflection in AB and BC respectively. The distance between the circumcenters of triangles ABC and P_(1) P_(2) is

Let ABC be a triangle in which AB = AC. Let L be the locus of points X inside or on the triangle such that BX = CX. Which of the following statements are correct? 1. L is a straight line passing through A and in-centre of triangle ABC is on L. 2. L is a straight line passing through A and orthocentre of triangle ABC is a point on L. 3. L is a straight line passing through A and centroid of triangle ABC is a point on L. Select the correct answer using the code given below.

Let P be the point inside that Delta ABC. Such that angle APB=angle BPC=angle CPA. Prove that PA+ PB +PC =sqrt((a^(2)+b^(2)+ c^(2))/(2)+ 2sqrt3Delta,)where a,b,c Delta are the sides and the area of Delta ABC.

Let A(5,-3), B (2,7) and C (-1, 2) be the vertices of a triangle ABC . If P is a point inside the triangle ABC such that the triangle APC,APB and BPC have equal areas, then length of the line segment PB is: