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A rectangle, a circle, and a triangle are drawn on a plane

Puzzling Asked on July 3, 2021

A rectangle, a circle and a triangle are drawn on a plane. What is the maximum possible number of points of intersection? The sides of the triangle are not collinear with any of the sides of the rectangle.

4 Answers

Here is a proof that the answer by @Avi is the largest possible.

We have the following lemma, which is intuitively clear and also can be proved rigorously:

With the lemma, we now see that

and adding them together gives the maximum number.

Correct answer by WhatsUp on July 3, 2021

Answered by Avi on July 3, 2021

Answered by Swiss Frank on July 3, 2021

Draw a rectangle. Use the point of the intersection of the two diagonals as centre and and draw a circle with radius of length less than the distance from the intersection of the diagonals to one of the vertices, as shown. Then draw a triangle as shown. The number of points of intersection of the rectangle, the circle and the triangle is twenty.

Oct17

Answered by Vassilis Parassidis on July 3, 2021

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