A Circle and the Amazing Quadrilateral

Geometry Level 5

A circle with radius 2 cm intersects (exactly) once on each of the sides and diagonals (as line segments) of a quadrilateral at their midpoints. What is the area of the quadrilateral (up to hundredths place)?

Treat the sides and diagonals as line segments.

The answer is 13.86.

2 solutions

Michael Mendrin
Jul 5, 2014

The trouble with this problem is that the solution is not unique. There's an infinity of possible quadrilaterals that have this property. You need to reword your problem so that it has an unique solution.

Indeed, the area of quadrilateral $ABCD$ is not uniquely determined. Here is a simple construction: Let $P$ , $Q$ , and $R$ be any points on the circle. Construct points $A$ , $B$ , and $C$ such that $P$ , $Q$ , and $R$ are the midpoints of the sides of triangle $ABC$ , as shown. Let $D$ be the orthocenter of triangle $ABC$ .

https://i.imgur.com/Tjr4nEk.png

Then the circle is the nine-point circle of triangle $ABC$ , so it passes through the midpoints of the sides and diagonals of quadrilateral $ABCD$ . By playing with the diagram, you can easily vary the area of quadrilateral $ABCD$ .

Jon Haussmann - 6 years, 11 months ago

Thanks for the correction, sir.

Ryan Redz - 6 years, 11 months ago

Thanks for the correction, sir.

Ryan Redz - 6 years, 11 months ago

Ryan Redz
Jun 24, 2014

Construct a circle with center C inscribed in an equilateral triangle ABD. Construct segments AC, BC and CD. the quadrilateral is actually the concave quadrilateral ABCD. From there, you can solve for the Area which is approx. = 13.86.

I've edited your question slightly for more clarity.

Why is this construction unique?

Calvin Lin Staff - 6 years, 11 months ago

Thanks for the correction, sir.

Ryan Redz - 6 years, 11 months ago

A Circle and the Amazing Quadrilateral

The answer is 13.86.

2 solutions

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