Question

Prove that a quadrilateral formed by joining the incentres of 4 triangles formed by diagonals of a cyclic quadrilateral is rectangle?

Answer 1

Answer:

The figure below is the same as above, except with the points J,K,L, M labelled and the line DB added. By definition J,K,L,M are the midpoints of their respective sides.

 

1  JM is the midsegment of the triangle ABD  The midsegment of a triangle is a line linking the midpoints of two sides (See Midsegment of a triangle)

2  JM is half DB and parallel to it  From the properties of the midsegment of a triangle

3  Likewise in triangle DBC, LK is also half DB and parallel to it  From the properties of the midsegment of a triangle

4  JKLM is a parallelogram  A pair of opposite sides (LK and JM) are parallel and congruent