Tangential quadrilateral

Question

I just proved that "In every tangent quadrilateral the sums of the lengths of opposite sides are qual. Conversely, every quadrilateral with this property is a tangent quadrilateral".

Now I am a little bit confused with the following property: "There are three lines thorugh a point X and three lines thorugh a point Y, and the intersections build a (distorted) 3 times 3 grid. Show that if two diagonal quadrilaterals are tangential quadrilaterals in this grid => then the big quadrilateral is also a tangential quadrilateral"

Do you have any suggestions? I have problems visualizing this text.

Answer 1

look at AFCD: AD+CF=CD+AF look at ABCF: AF+BC=CF+AB

so you get: AD+CF+AF+BC=CD+AF+CF+AB

that is : AD+BC=CD+AB, you already proof the first part. so ABCD is the tangential quadrilateral.

if you want to draw the circle, simply make the bisectors of A and C, they cross at L, the L is the center of the circle.

Answer 2

I have problems visualizing this text.

To help your visualization, here is an image, created with Cinderella.