In a convex quadrilateral A B C D, we have points P ∈ A B ; R ∈ A D. Lines P D and R B intersect at a point S. Assume we can inscribe a circle in each of the quadrilaterals A P S R and C D S B. How do I prove that we can inscribe a circle in A B C D

Question

In a convex quadrilateral   A  B  C  D, we have points   P  ∈  A  B  ;  R  ∈  A  D. Lines   P  D and   R  B intersect at a point   S. Assume we can inscribe a circle in each of the quadrilaterals   A  P  S  R and   C  D  S  B. How do I prove that we can inscribe a circle in   A  B  C  D

Olivia Petersen · Accepted Answer

Hint: Simply mark the four points of tangeny of the circle inscribed in   A  P  S  R and the other four points of tangeny of the circle inscribed in   C  D  S  B. Then assign edge-length variables to the segments formed by adding these four points to the rest of the points given to you (you get a lot of segments). Have in mind that the tangents to the same circle from a given point are equal! Also, notice that that the two interior tangents to the two incircles are equal (the ones through the point   S)! Then carefully form all the equations (sums of lengths) for these edge-length variables and after simplifying them you will obtain at the end that   A  B  +  C  D  =  B  C  +  C  D which is a necessary and sufficient condition for the existence of an incircle in   A  B  C  D.

n a convex quadrilateral ABCD, we have points P in AB;R in AD. Lines PD and RB intersect at a point S. Assume we can inscribe a circle in each of the quadrilaterals APSR and CDSB. How do I prove that we can inscribe a circle in ABCD

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