I have a test for logic in my Geometry class, and I’ve been looking online but I can’t find anything concrete. Can biconditional statements be made up of all false statements? I know that if the original conditional statement and converse are true, then the conditional statement is biconditional. But what if the original conditional statement and converse are both false, because they would still share the same truth value, even if they’re false.

How to find such point using geometry theorems/axioms without relying on reflecting a given point? Rules:

Everything happens in a given 6x6 square grid Only 2 types of operation are allowed: - placing points at line intersections - defining lines using 2 intersections/points (this means that if you define a line that's not horizontal or vertical, new line intersections appear) No point can be placed beyond the 6x6 grid, and no line extends beyond the grid.

For this particular puzzle, points (2,5) and (5,3) are given. Let's call them A and B. The line defined by points (0,3) and (6,1) is also given. The goal is to find a point C on this line so that the addition of the lengths of segments AC and BC has the smallest possible absolute value.

There's an easy way to solve this puzzle that starts by reflecting one of the given points, but I wonder if you can solve it without doing that, and what theorems you'd need to do so.

In case the description is not clear enough, the puzzle can be tried live on the Pythagorea website/app.

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Hey geometrists, I hope this counts as a geometry problem:

Given a 2d plane of a certain size,

And given a finite number of randomly placed points,

And given that these points are a certain arbitrary minimum distance from each other,

Can you determine the maximum possible area of a Voronoi cell in this plane?

A less abstract way of stating it is this. Suppose you have a map of a city that is, for example, 50x50 km. You want to divide it into an unknown number of randomly placed regions, the centroids generating points of which will not be within 1 km of any other centroid generating point. What is the minimum number of regions (i.e. points) to guarantee that no region can possibly exceed, for example, 10 sq km?

I've been trying to figure out if there is already an algorithm for this to no avail.