What is the area of quadrilateral?

Question

The incircle of a quadrilateral of preimeter 2P has radius r. What is the area of quadrilateral. 

Answer 1

Draw the line segments from the center of the circle to each of the 4 vertices of the quadrilateral.  This divides the quadrilateral into 4 triangles.  One of the sides of each triangle is a side of the quadrilateral.  And the area of the quadrilateral is the sum of the areas of all 4 triangles.

 

Now pick one of the sides of the quadrilateral and draw the line segment from the center of the circle to the point where the circle is tangent to that side.  This line will be perpendicular to the side, because the radius of a circle is always perpendicular to any tangent line.  That means that this line constitutes the height of the triangle that contains this line.  And the base of that triangle is the side of the quadrilateral that you drew the line to.  So that area of this triangle is equal to 1/2 r times the length of that side.  The same is true for all four of the triangles you divided the quadrilateral into at the beginning.

 

So the area of the quadrilateral is equal to 1/2 r s₁ + 1/2 r s₂ + 1/2 r s₃ + 1/2 r s₄, where s₁, s₂, s₃ and s₄
are the sides of the quadrilateral. Factor out 1/2 r and this is equal to 1/2 r (s₁ + s₂ + s₃ + s₄).

 

But s₁ + s₂ + s₃ + s₄ is the perimeter of the quadrilateral, which is 2P.  So this is equal to 1/2 r (2P) = rP.

 

So the area of the quadrilateral is rP.