Proof of foot of perpendicular, image and length of perpendicular

 Proof of foot of perpendicular of a point to a line, image of a point with respect to line and length of perpendicular from a point to a line.

Proof of Derangement formula

 Derangement Formula

Inclusion exclusion principle

Proof of perpendicular distance of a point from a line

 Perpendicular distance of a point from a line

Position of two point with respect to a line

Position of two point with respect to a line

Proof Images of orthocenter in any side of triangle lie on circumcircle

 Proof Images of orthocenter in any side of triangle lie on circumcircle

Proof of coordinates of orthocenter of triangle

 Proof of coordinates of orthocenter of triangle

Proof of coordinates of circumcenter of triangle

 Proof of coordinates of circumcenter of triangle

Proof of coordinates of excenter

 

Proof of coordinates of Incenter

 Coordinates of Incenter

To know the proof of BD:DC::c:b visit following link

https://maths49.blogspot.com/2022/01/angle-bisector-divides-opposite-side-in.html

Or 

https://maths49.blogspot.com/2022/01/internal-angle-bisector-divides.html

To know the proof of AI:ID::b+c:a visit following link

https://maths49.blogspot.com/2022/01/proof-of-incenter-divides-internal.html

Proof of Incenter divides the internal angle bisector in the ratio of b+c:a

 Incenter of triangle divides the internal angle bisector in ratio of b+c:a

For proof of BD:CD::c:b visit following links

https://maths49.blogspot.com/2022/01/angle-bisector-divides-opposite-side-in.html

Or 

https://maths49.blogspot.com/2022/01/internal-angle-bisector-divides.html